How many barrels (40gal) fit in a standard bag of holding? 

You will need to know the height and width of the barrel. These vary. (the bag's inner space is 4 ft deep. The opening can be up to 2 ft wide.)

You need to know the weight of the barrel which can vary by materials and content. (the maximum weight is 500 lbs)

You need to know the volume of space a barrel takes up. (the maximum volume is 64 cubic feet)

Exceeding the restrictions of the bag's inner space causes it to rip - resulting in its immediate destruction and the contents being cast into the Astral Plane.

If you're just talking about how much liquid - it is 64 cubic feet, or 478.753 US gallons / 398.645 Imperial Gallons.

Quote from J_RA_theBeetleBlah >>

How many barrels (40gal) fit in a standard bag of holding? 

 One. A bag of holding can hold up to 500 pounds, and Google says a full 40 gal. barrel is about 434 lbs (and is about 23.8" Dia. x 30.1" H, and weights about 100 lbs empty)

Official Cask Units (English)

Type     Gallons      Litres      Tun
Tun        252         954         1
Butt       126         477         1/2
Puncheon    84         318         1/3
Hogshead    63         238         1/4
Tierce      42         159         1/6
Barrel      32         118         1/8
Rundlet     18          68         1/14
Kilderkin   16          59         1/16
Firkin       8          30         1/32
Pin          4          15         1/64

edit: found a pic here: casks

I feel I know where this question is ultimately headed, so I'll cut to the chase: a bag of holding can hold just over 59 U.S. liquid gallons of ale. So, cut out the barrel and just dip your mug in -- no need to over complicate it!

For reference, the 64 cubic feet of the bag of holding equates to about 478.8 U.S. liquid gallons. However, with the 500 lb. weight maximum, ale (and beer) which weighs about the same as water, reaches that limit far before it is filled.

Quote from Vansen >>

For reference, the 64 cubic feet of the bag of holding equates to about 478.8 U.S. liquid gallons. However, with the 500 lb. weight maximum, ale (and beer) which weighs about the same as water, reaches that limit far before it is filled.

 Spot on.

also 478.8 gallons weighs 3995.8 lbs

Water density is 997 kg/m³ - Beer density is only slightly higher at 1030 kg/m³. 

500 lbs of water equals 59.9 US Gallons (gal) and would use up 8 cubic feet of space - beer would be slightly lower

There are actual barrel stats in the PHB, Chapter 5.

Container Capacity

Container	Capacity
Backpack*	1 cubic foot/30 pounds of gear
Barrel	40 gallons liquid, 4 cubic feet solid
Basket	2 cubic feet/40 pounds of gear
Bottle	1 1/2 pints liquid
Bucket	3 gallons liquid, 1/2 cubic foot solid
Chest	12 cubic feet/300 pounds gear
Flask or tankard	1 pint liquid
Jug or pitcher	1 gallon liquid
Pot, iron	1 gallon liquid
Pouch	1/5 cubic foot/6 pounds of gear
Sack	1 cubic foot/30 pounds of gear
Vial	4 ounces liquid
Waterskin	4 pints liquid

 and 

Other Adventuring Gear

Item	Cost	Weight
...	...	...
Barrel	2 gp	70 lb.

And I suppose while I'm at it, rules for barrel hardness from DMG, Chapter 8:

Object Armor Class

Substance	AC	Substance	AC
Cloth, paper, rope	11	Iron, steel	19
Crystal, glass, ice	13	Mithral	21
Wood, bone	15	Adamantine	23
Stone	17

...

Object Hit Points

Size	Fragile	Resilient
Tiny (bottle, lock)	2 (1d4)	5 (2d4)
Small (chest, lute)	3 (1d6)	10 (3d6)
Medium (barrel, chandelier)	4 (1d8)	18 (4d8)
Large (cart, 10-ft.-by-10-ft. window)	5 (1d10)	27 (5d10)

Objects and Damage Types. Objects are immune to poison and psychic damage. You might decide that some damage types are more effective against a particular object or substance than others. For example, bludgeoning damage works well for smashing things but not for cutting through rope or leather. Paper or cloth objects might be vulnerable to fire and lightning damage. A pick can chip away stone but can’t effectively cut down a tree. As always, use your best judgment.

Damage Threshold. Big objects such as castle walls often have extra resilience represented by a damage threshold. An object with a damage threshold has immunity to all damage unless it takes an amount of damage from a single attack or effect equal to or greater than its damage threshold, in which case it takes damage as normal. Any damage that fails to meet or exceed the object’s damage threshold is considered superficial and doesn’t reduce the object’s hit points.