One of the players in my campaign is a Goliath with a brewers kit. He makes barrels of ale and carries it around to offer people. After some math it was discovered that a 32 gallon of ale is like 266 cups. I'm aware that Goliath have the powerful build trait, but I feel as though there should be some restrictions to  carrying a barrel around. Any suggestions or ways to go about it? 

Just use the carrying capacity and encumbrance rules. If their strength is high enough to lift 400 pounds, great. If not, too bad.

One US gallon of water is 8,35 pounds and therefore 40 gallons weigh 267.2 pounds (not including the weight of the barrel itself, which according the the phb is 70lbs empty...so that's 337.2lbs fully filled.

Its some serious dent in his carry capacity once other equipment is taken into account.

Even if he can carry it, I'm wo seeing if having a barrel on his character would hinder his movement /combat ability. 

Ooooh time for some Math!

I'd also rule as it is such a large object you need two hands to carry it, so you have to keep putting it down/picking it up and as per the encumbrance variant rule in the PHB (pg176):

"If you carry weight in excess of 5 times your Strength score, you are encumbered, which means your speed drops by 10ft.
If you carry weight in excess of 10 times your Strength score, up to your maximum carrying capacity, you are instead heavily encumbered, which means your speed drops by 20ft and you have disadvantage on ability checks, attack rolls, and saving throws that use Strength, Dexterity, or Constitution."

So assuming the Goliath had Str 20, that means for a Goliath his overall carrying capacity is : 15x20 =300, +1 size category makes it 600lb total carrying capacity.

At 200lbs the Goliath speed is reduced by 10ft (would be 100lbs normally but you double the total for being a Goliath)

At 400lbs the Goliath speed is reduced by 20ft and have disadvantage on all ability checks, attack rolls, and saving throws that use Strength, Dexterity, or Constitution.

Barrel of Ale = 337lbs (rounded down)

Assuming Base Barbarian equipment (assuming Great axe 7lbs, two hand axes 4lbs, explorer's pack 69lbs and four javelins 8lbs) = 88lbs (explorers pack breaks down to: backpack 5lbs, bedroll 7lbs, mess kit 1lb, tinderbox 1lb,10x torches 10lbs, 10 days of rations 20lbs, water skin 5lbs, 50ft hempen rope 10lbs)

Assuming Out-lander Background starting equipment: (A staff 4lbs, a hunting trap 25lbs, a trophy from an animal you killed 0 lbs, traveler's clothes 4lbs, and a belt pouch 1lb) = 34lbs (I've not put a weight for the trophy as it could be a claw or fang which is negligible weight).

Grand total so far is: 337 + 88 + 34 = 459lbs and would render the Barbarian at heavily encumbered.

Due to various things you'd have to home brew I'd not let them use the barrel as weapon but say it has AC12 & 20hp in case they try to use it for cover, but any damage it takes results in a leak and loss of beer/ale etc, each point of damage would equal 5% loss of liquid, or 13lb of weight lost from the Barrel (rounding down) to a minimum of 70lbs once the barrel has lost all its contents.

Rather handily the number of cups you mentioned in your first post means each cup removes about 1lb of liquid (again rounded down).

Hope that helps....

Rob did an excellent job of covering the math. So I'd like to mention the situational hindrances that the goliath's keg toting would cause:

The character will probably want to carry the keg on one shoulder, like a tough guy. Well, that means he would be effectively blind to that side, since he can't see through the cask. So that means NO passive perception on that side, NO (or at best disadvantage on) active perception checks on that side, and an enemy could quite easily surprise him from that side. All it may take is one well placed rogue to remind the goliath of the effectiveness of Sneak Attack damage. 

Also... you can't fight with a cask of ale in your arms. So, whenever combat begins, the goliath won't be able to attack in the first round. Instead he'll have to use an action to put the cask down safely. Sure, he could just drop it as a bonus action, but that would risk damage to the cask and a possible loss of the contents. 

Of course, the goliath could simply buy a mule or a clydesdale to carry his cask for him. Rob's math puts the full cask at 337 pounds, which is right in the middle of a mule's medium-encumbered carrying capacity. But the mule might not be able to negotiate a dungeon very well. 

Ever since working through this idea, I've had this image in mind...

One of those casks has about 1 liter of capacity so would make a much more reasonable item to cart around, weighs about 2lbs (rounded down) when empty and gives 4 cups/2 pints of liquid so full would weigh 4lbs full filled up...although wearing  it around your neck may not be practical it is a fashion statement!

You could have a couple of these on a back pack style back harness and they would fit in a bag of holding so they could be a good work around.

This is definitely a case of like... either buy a smaller barrel, or buy some leather straps and wear that thing like a backpack.

I think he's been trying to justify that the barrel goes into his backpack. I think he's trying to say that he is so large that his backpack is big enough to hold the barrel as well.

That might be reasonable, if he gets a custom backpack, because even one that was made for a goliath is probably not going to be the right shape to hold a barrel.

Well. Let's revisit the math.

In modern terms, a half-barrel is 15.5 gallons and is 24" tall and 17" in diameter. That's a volume of 5447 cu in. for 15.5 gallons. So 32 gallons would be 11246 cubic inches. That's a cylinder about 33" tall and 21" in diameter. If a goliath is 7'6" tall, I think it is reasonable that a keg of that size would fit in a specially-modified backback. 

BUT! The weight of the keg would still put the character well into the heavily encumbered range. AND that's if the character carries ONLY the keg! Who's going to carry his other stuff? All the other characters will have to divide up the rest of the goliath's gear, possibly pushing several of them into the heavily encumbered category as well. 

Also... you didn't mention what level the characters are. Usually by the time a party hits tier two (around level 5 or 6) they have usually acquired a place to call home. Sure, they are usually out adventuring, but they at least have a place to return to. Maybe a house, or a small tower in a keep, or a farm somewhere. 

So if the players are at that range, or will be soon, I think it's best to explain to the player that their character has the brewery set up at the house, and can brew to their heart's content as a downtime activity. But when it's time to go kill stuff, just take what you need to kill stuff.