As it stands, making any large-sized playable race in D&D 5E is more of a problem than is likely to be worthwhile. According to the DMG, a large-size playable race would deal double weapon damage at level 1, and with the way hit dice work in the MM, it could be argued that their class hit dice would be upgraded by one die type, meaning a large-sized fighter for example would have d12 hit dice instead of d10 due to size.
These huge advantages would be balanced out a bit by the fact that a large PC would have to squeeze in a lot of common situations – traveling through Dwarven tunnels or visiting the ubiquitous pseudo-Medieval taverns. I’d assume, though, that the DM would just have to adjust for that, reducing the number of five-foot-wide corridors and so on in a given adventure, or else the player playing the large PC would just be left out. Somewhat balanced, but definitely no fun, leaving a situation where the PC would have all the advantages and probably few, or none, of the constraints of being large.
The effects of the Enlarge/Reduce spell in the PHB suggest another interpretation, a bit less advantageous than what the DMG and MM imply. An enlarged creature deals +1d4 damage with their enlarged weapon and have advantage on Strength checks and Strength saves and that’s pretty much it. Presumably, the DM just improvises the effects of being enlarged where it would be a detriment rather than an advantage, and obviously a savvy caster would not enlarge an ally in the middle of a cramped room or hallway designed for medium-sized species.
I don’t think either approach to a large-sized playable race is particularly good, whether taking our cue from the DMG and MM, or from the PHB. That being said, I like the idea of a large-sized playable race a lot. I think it adds something to a setting and to the options available to players, and there should be a way to balance things out. In 3.X this balance came in part with a penalty to Armor Class and stealth checks, and I think that makes sense conceptually.
So here is what I think a large-sized race or species in D&D 5E should include: +1d4 damage from large-sized weapons, advantage on Strength saves and Strength checks, disadvantage on Dexterity saves and Dexterity checks, +1 hit points per level, and a cost of living multiplied by four (including meals, water skins, clothing, equipment, etc.).
An Example: Dark Sun’s Half Giant
Ability Score Increase. Your Strength and Constitution score both increase by 2.
Age. Half-giants live about twice as long as humans, becoming adults around the age of 25 and often living to 170 (for the few who die of old age).
Alignment. Half-giants adopt their alignment from the people they spend the most time with, or fear or respect most. This means that their alignment will be more subject to change than others, though one axis till tend to remain consistent. So they might be consistently Good, but sometimes Chaotic and sometimes Lawful, or consistently Chaotic, but sometimes Good and sometimes Evil.
Powerful Build. Half-giants have advantage on Strength checks and Strength saves. They also have disadvantage on Dexterity checks and Dexterity saves. In addition, their build grants them +1 hit points per level.
Size. Half-giants are Large sized creatures. They occupy a 10′ by 10′ square, and have a 5′ reach. They also deal +1d4 damage with all weapons, in addition to the listed damage.
Speed. Half-giants have a base speed of 35 feet.
Upkeep. The cost of living and all equipment for a half-giant costs four times the usual amount.