If you insist on standing or sitting upright, the blood in your head would try to fall towards your feet, and if you didn't take counter-measures you'd probably black out around 5 G, due to falling blood pressure in your head reducing the oxygen supply to your brain. Lying down solves this problem. The only minor inconvenience is that if you want to lift your arms they're very heavy.
The main reason why astronauts and cosmonauts go through such exhaustive selection and training is because their flights are very expensive. And so if a problem arose, like they got appendicitis, and the mission had to be abandoned, this would be a terrible waste of (taxpayers') money. The early astronauts were test pilots, because no-one knew what they were going to experience, and they were flying in dangerous vehicles. But today, essentially anyone who's in reasonable health could go and stay in orbit - if there was an economical launch vehicle available. Certainly anyone who would fly in an aeroplane or ride on a roller coaster will have no problem.
That isn't to say there aren't a number of health "issues". We needn't call them "problems" any more because we know the answers - at least enough to allow us to operate a safe space tourism business. These two issues are "space sickness" and the long-term effects of living in space.
Although staying in space for a short time is quite safe, it's well known that many people who visit space, on the US space shuttle or the Russian "Soyuz" launcher, feel sick, and even vomit. The reason for this is much the same as the reason why some people are sick in cars or ships - "motion sickness" or "travel sickness". (For those who are interested in the details, it's due to the conflict between what your eyes are telling your brain and what your inner ear is telling your brain about your bodily position. Inside a ship your eyes tell you you're not moving, but your inner ears tell you you are. That's why going out on deck helps - your eyes can see you are moving.)
For a long time it was said that "space sickness" is somehow special, and different from other forms of travel sickness. But it isn't. Sure, the precise type of movement of the inner ear is a little different in zero G - just as car-sickness, sea-sickness and air-sickness are a little different from each other. But the good thing is that normal travel sickness medicines are entirely effective against "space sickness", as they are against other forms of motion sickness.
The main reason people are sick on the space shuttle is that they are asked not to take anti-sickness medication, so that NASA staff can do research on the phenomenon! It is indeed possible to do "research" on space sickness - just as it would be possible to do research on sea sickness. But there's no need to. Brain scientists don't fully "understand" sea-sickness - but the cruise industry doesn't pay them to do research. They just make it easy for passengers to get hold of effective medication! The commercial approach - low cost and effective. The same will be done for space tourism, and no-one will need to be sick. Unfortunately, most journalists still don't know this, and trot out the same old line "It's a pity that everyone's going to be sick, so it'll never get very popular..."
One phenomenon - it can't really be called a problem - is the fact that in zero gravity, as the fluids in your body are no longer pulled down towards your feet by gravity, it tends to accumulate higher up in your body, creating what some people call "bird legs" - the muscles of your legs appear smaller than on Earth - and also "fat-face" (or perhaps more appropriately "Moon-face"). Your family and friends will look a bit different in zero G! (You can see this in photos of astronauts.)
There's a serious side to it which is that this also gives you a slight feeling of nasal congestion which may be uncomfortable for some people. So using decongestants may be popular. In addition, having a head cold may be more unpleasant than on Earth - so medication to suppress the symptoms (now common on Earth) may be desirable.
Unfortunately astronauts and cosmonauts don't go to space if they have a cold - so we don't have good information on this! This is an example of how the experiments desirable for space tourism are different from the experiments that space agencies do aimed at long term space flight, etc. As they get wise, maybe you'll see an advertisement saying "Astronauts wanted - must have streaming cold".
On Earth your muscles and bones are continually subject to gravity. This exerts forces on them, generally putting the muscles in tension and the bones in compression. When you're floating in zero-G most of these forces disappear, and your muscles are very relaxed. (One result of this is that people living in orbit don't get very tired, and tend to sleep less than down here - 3 - 4 hours is said to be common.) This condition is no problem for a week or so, but if it continues for months, the muscles start to shrink through lack of use. (When you train a lot, your muscles grow in volume; when you don't use them they shrink!) This includes the muscles of the heart, because pumping blood in zero G is much easier than in 1 G. So astronauts and cosmonauts staying in orbit for months make time for daily physical exercise to keep their muscle strength up so that they won't be weak when they get back to Earth.
Another effect of living in weightlessness is that, like your muscles, your bones also lose mass. Though bones seem pretty solid, they're alive, and like many biological systems they're in equilibrium between building themselves up and breaking themselves down. So they continually take calcium from the blood and release calcium into the blood. But when the stresses on the bones fall, as when you're floating around in zero G, less calcium is taken up from the blood because the cells sense less need for it. For a week or so this is no problem. But if it continued for months, your bones would get significantly weaker. So people who spend long periods in space have to use special trousers with strong elastic joining the belt to the ankles, which puts compressive forces on the leg bones and joints, and convinces them to take up more calcium. It may not be so elegant, but by doing this they don't lose so much bone mass. Further research will probably develop even better remedies. But we have to select priorities. It's much more urgent to get launch costs down - by developing a space tourism industry, among other things - than to do this research soon. Current knowledge is quite adequate for building a medically safe orbital tourism industry - and after that it will be much cheaper to do this research. That would be much better than continuing to spend hundreds of $millions to perfect this treatment when there's no urgent need for it.
Nuclear radiation, including both high-energy particles such as protons and other atomic nuclei, and high-frequency electro-magnetic waves, X-rays and Gamma-rays, can damage biological cells by ionizing molecules inside them. Most of the time this damage is simply repaired automatically. But if a lot of radiation is absorbed, the damage can overcome the body's ability to repair it, and the damaged cells may die or may become cancerous (that is, growing uncontrollably).
In space, all of these types of radiation are present to some extent - protons and electrons trapped in the Earth's radiation belts and coming from the Sun, and energetic particles coming from further away ("cosmic radiation"). The Earth's atmosphere protects us from them to a large extent (but not completely; people living at high altitude receive considerably higher "background radiation" doses than those living lower down!) Roughly speaking, the longer you stay in space, and the higher the orbit you travel in, the more radiation you absorb, and so the higher the chance of damage. In most cases, spending a few days in orbit will pose no risk at all. The radiation absorbed will be equivalent to the average radiation you receive in a year on Earth (from X-rays, background radiation, and environmental pollution). There are long-established international standards for the recommended maximum radiation doses both for the general public, and for "radiation workers" - people working in hospitals who use nuclear materials, in the military, in nuclear power stations, and astronauts. None of the several hundred people who've been to space (some for months) have shown any signs of radiation sickness.
In addition to keeping within known health standards, one precaution will be necessary: there are occasional "storms" on the Sun, generating "solar flares" which give out much more radiation than usual for a few hours. Conveniently these flares can be seen minutes before the particles arrive at the Earth, and so people can take precautions. For example, if a Concorde is in flight at its supersonic cruising altitude of about 20 km when a solar flare occurs, it will reduce speed and reduce its altitude to travel deeper within the atmosphere, thereby protecting the passengers from receiving raised radiation levels. In the same way, orbital hotels will have a "storm shelter" - one or more parts of the hotel in which everyone can shelter from the radiation. The shielding needed is just thick walls, made of some material like metal, concrete or even paper, or just water tanks, since water is a good shield.
We know the above from the experience of the people who've already been to space. These include several hundred people who've been in orbit for a week or two, and a few tens who've been in orbit for a few months. As a result, sufficient research has been done on the effects of living in space to be able to offer safe space tourism services. Those who choose to work in space hotels and other businesses in space will be a new kind of "radiation worker". As such they'll receive more radiation than the general public. This will be at their choice; their health will be closely monitored; and the additional risks they will be taking will be very small. One proviso is that, depending on the facility in question, like radiation workers today, staff will generally only work there after having children, since it's particularly important to prevent damage to the reproductive system. Having babies in space will be a subject for future research!
We mustn't forget that there will also be health benefits of living in zero G. People with painful backs will get relief, as they can floating in a swimming pool. Poor blood circulation may well be improved. There's also a possibility that useful discoveries will be made, such as that some kinds of surgery will be easier or safer in zero gravity. So "space hospitals" may become a significant business. But that can happen only once launch costs are cheaper! They're not a priority for research today. However, we can envisage that some people may choose to spend long periods of time in orbit - perhaps in a hotel offering partial-gravity, or even in apartment-blocks!