How tall do you have to be to be an astronaut?

Straight from the NASA website, there are actually just a few requirements you need to meet to become an astronaut:

1. A Bachelor's degree in engineering, biological science, physical science, or mathematics.
2. Three years of related work experience
3. Ability to pass a NASA space physical, which includes: having vision 20/200 uncorrected or better, correctable to 20/20; blood pressure of 140/90 when sitting.
4. Height between 58.5 and 76 inches. (from 4 feet, 10.5 inches to 6 feet, 4 inches or 148.59cm to 193.04cm)

If you want to become a Commander or a Pilot, the requirements are a little more stringent. You have to have uncorrected vision of 20/100 or better (also correctable to 20/20), and a height between 62 and 75 inches.

More detailed information about how the selection process for astronauts works can be found on the Astronauts home page.

Qualification to become an astronaut isn't determined by ability to fit in a space suit, or at least it is unlikely to be a consideration for you. What matters is absence of any issues that would impair your ability to perform the tasks you are there to perform (and not become a liability to your crewmates), and, very importantly, the skills, knowledge, and discipline to do the job you are there to do under the circumstances it needs to be done.

If you're only going to be ready to apply in, say, 10-15 years from now, many things could be very different. The current fleet of space suits will probably be completely replaced by newer designs with a wider range of accommodation. However, some things are less flexible and less likely to become more accommodating. A spacecraft pilot station, like the driver's seat of a car or the pilot's seat of an aircraft is going to be constrained in just how wide of a range of adjustment it can offer its occupant. Nevertheless, it seems like you would have no issue in this regard.

As other answers have already explained, the basic height requirement is 4 feet, 10.5 inches to 6 feet, 4 inches.

However, just meeting that (and the other basic specifications) isn't going to get you far. I actually applied for the NASA candidate program several years back (2012 or so?), shortly after I'd met all of the listed requirements (they have to pick someone, right?). I got a brief response from NASA a couple months later (heavily paraphrased):

"Thanks for your application, you are one of 2500+ applicants nationwide whose resume didn't go straight in the trash for not even meeting the requirements. Please don't send follow up letters, or expect any further response unless you are one of the 50-ish candidates we short-list to interview in person for the 15 crew and back-up crew positions we expect to be available in 3-4 years."
_{I may still have the letter somewhere if people really want to see the exact phrasing?}

If this is something you really, really are willing to dedicate your life to, look at the typical profiles of past crew members for reference and try to acquire some additional skills above and beyond the short list NASA gives. Based on my own research and on some things mentioned in the application brochure, some of your options for standing out include (but certainly aren't limited to):

• Become a military pilot (to be able to land the return craft if needed)
• Get (and stay) in top physical shape (no need to body build, just all-around endurance)
• Get 1 or more advanced degrees in aerospace-related engineering/technology fields (understanding and practicing all the different fields involved)
• Teach/work in an engineering/technology field for a few years
• Learn Russian (lots of interaction with the Russian program for the foreseeable future)
• Get scuba certified and experienced (micro-gravity activities in awkward gear)
• Get EMT certified (medical care in emergency situations)

Even with all of the above though, there just aren't that many spots on the NASA program. However commercial programs (SpaceX, Blue Horizon, etc) are reasonably likely to have manned programs by the time you are old enough to participate, and are likely to have similar needs/requirements. But even if you never did get into space (I doubt I will at this point), there are many ways to be involved in space exploration, and any and all preparation you may make to be an astronaut will open some amazing career opportunities to you in any field you decide to pursue.

Remember that the first round of NASA astronauts were all experienced test pilots and aviators who already had to meet a stringent range of physical and mental requirements. It's hard to build aircraft and spacecraft that can accomodate more than a few inches' difference in height and width (at least not without a mass penalty).

The Mercury, Gemini, and Apollo spacecraft were tiny - Mercury astronauts didn't sit in the capsule as much as wear it, and the Apollo command module wasn't much bigger than a modern SUV in terms of interior volume.

Requirements were relaxed as more civilians joined the astronaut corps, but there will still be limits on size and general health. The ride uphill puts a lot of stress on the body¹, and because mass budgets will always be tight, seats and controls are going to be fixed at "one size fits most".

Having said all that...

As long as you take care of yourself, eat properly, move around enough to get your heart working every day, give your eyes a break (meaning, don't spend all your free time in front of a screen reading tiny, tiny text) and take advantage of every educational opportunity that comes your way, you shouldn't have any problems being accepted into the astronaut corps².