Okay. Modern horsemen use a simple formula. The combined weight of rider and gear should not exceed 20 percent of the weight of the horse. So, our 300 pound rider would need at least a 1600 pound horse, if riding English, closer to 1700 if he'd prefer a western saddle. (Western saddles weigh around twice as much as English ones). Horses that big certainly do exist, but most of them are draft horses - draft horses are built to pull, not carry, and need to have a slightly lighter rider than a light horse of the same weight.
Now, 20 percent is a rough guideline. Here are a few things you should also consider when having your characters mount up...or load up...a horse.
1. Donkeys can carry more than horses. Mules can carry slightly more, but their capacity is generally closer to a horse than a donkey. (The same goes for hinnies).
2. A horse can carry more weight of active, cooperating rider, than "dead weight". If you're asking your horse to carry a pack, a corpse, a wounded man, or somebody with no clue how to ride - then that will be an effectively higher weight. Beginner riders are the hardest thing for a horse to carry because they often work actively against the horse rather than with it. A highly skilled rider can ride a mount considerably smaller - as when an adult gets on a small pony to train it.
3. Ponies can carry more weight, in proportion to their size, than horses. Ponies of the tough northern European breeds - and I include the Icelandic horse in this even though it's not generally considered a pony - have amazing carrying capacity. A 12.2 Icelandic easily carried my 130 pounds for 5 hours in perfect comfort for both of us. Dartmoor and Exmoor ponies are well known for being able to be ridden by full-grown men.
4. Obviously, an out of shape horse can carry less than a fit one. Older horses and younger ones (<5 years old) also need care taken about the weight they're asked to carry.
Now, one thought that might come out of all of this - what about a knight's charger? First of all, plate armor weighs a lot less than people think. It weighed from 45 to 65 pounds. The armor on the horse probably, thus, adds another 80 pounds or so. People were smaller back then (but so were horses). Assuming we aren't talking about Gregor Clegane here, I'd estimate that a charger would be carrying about three hundred and fifty pounds. In comparison, a silver parade saddle weighs about as much as that horse armor - 75 to 100 pounds. Parade horses are specially conditioned to carry the weight and not asked to do so for very long.
The same was true of destriers - they were ridden only for tournaments, battle, and training. The knight would, the rest of the time, ride a traveling horse...a palfrey. His squire(s) would ride a lower quality horse or, possibly, a pony, and lead the war horse and a pack horse that would carry the armor. The war horse was, thus, specially conditioned to carry the weight...and not asked to do so for very long.
Now, obviously, you can break all of these rules if your characters wouldn't care about keeping their horse's backs in good condition. Or if it's an emergency...they have to get to the Baron's castle tonight and they only have one horse...
As for the second part of Doug's question - most riding establishments will not take riders over 200 pounds. Some may go up to 250. (Some are even lower). This especially goes for beginners. I suggest calling around the barns in your local area, finding the one with the highest weight limit and setting that as your target.
Image: A Quarter Horse gelding carrying a modern pack saddle.