It hasn't come up yet, but if one of my players wanted to have a large character it would be pretty easy for me to do. Everyone gets a feat at level 1 to help define their character, so a large character would take the "Large Character" feat, which has a prereq of being a race that should be large (eg, firbolg, goliath) and a minimum strength of 17.
Everyone gets all worked up over balance, but that's just nonsense. So what if there's an extra die on your melee hits? As DM, it is a simple matter to make certain that there are no Large magical weapons with damage bonuses, which most weapon-focused classes would be looking for. Most Large characters will probably wind up using a Medium 2-hander as a 1 handed weapon, meaning that you're dealing with either a +2 to AC for a shield or an offhand attack as a bonus action, neither of which should throw your game balance off by very much at all. Price your armor as barding, or multiply the cost by 4. Food consumption x 4 as well.
A large character can go most places, but not comfortably (see "squeezing" rules.) Note that the dwarves and Gandalf all had a good time in Bag End, but Gandalf was at constant risk of banging his head. While it is true that you can threaten more squares as a Large creature, it is also true that a whole lot of creatures can flank you.
Playing as a Large creature isn't any more of a "big deal" than playing as a small creature, really. There are some advantages, and some drawbacks... like any other significant aspect of a character. If your player wants to try that out, freakin' let him. Be prepared to offer an alternative if he decides he doesn't enjoy being a big oaf, but be willing to make accomodations so that he at least has the chance to try it out. There are plenty of people who have played characters with much more severe limitations and had a lot of fun doing it, so don't be afraid to let your player try a big character. I'd suggest starting without any serious penalties and seeing if the Large character needed to be reigned in, and not starting with a bunch of negatives that you might consider lifting later. For example, making someone give up an extra attack feature just seems extreme, to me. If your Biggun is overpowering everything and dramatically outshining other melee spec characters, you can always add a nerf later on, but do so as a group rather than dictating from on high.
But hey, that's just my opinion, man. Ultimately you need to do what will work for your players.