I have a 4' Celeste in a 5 gallon nursery pot. I live in western NY, so my fig comes inside in the winter. This year the roots grew out of the bottom of the container. What size containers does everyone suggest?

The bigger the better.   I have my large figs that are not in ground in 22 and 15gal. 
Keep in mind that you will need to move them for winter protection and that you only have limited space. 
Many fig growers with limited space have settled on 5 gal containers and root prune as required to maintain the fig.

Most of my figs are in 7 gal pots but the better ones get bigger containers.  I let my figs root into the ground every year then cut or pull up those roots when I bring my figs in for the winter.

rcantor, do you have any videos of you doing this?  or photos? I would love to see this being done. How tall are they?

Sorry have never weighed them.   They are easy enough to move around with a dolly.  You need 2 people to lift unto a truck.

weight order of magnitude;
22gals of water is approx. 88l * 1kg/L = 88KG (193 lbs).  A fig container would be somewhat less.

I like using fabric bags. They root prune the plants. This allows for more roots. It is said it makes say a 10 gallon fabric bag comparable to a 20 gallon pot. The roots when they hit the side get air and stop growing. They will never swirl. One could still root prune them occasionally. Mine are so new I have not yet done that. I'm fairly new to figs. Having fun with them. they seem fairly easy to grow.
I use root pouches as they are cheap! If you buy 10 10 gallon, 4 bucks  each.
http://www.greenhousemegastore.com/product/root-pouch-boxer-brown-fabric-pot-longest-lifespan/growing-pouches-and-bags

I use 20 and 25 gallon nursery pots for my older trees and 5 gallon nursery pots for everything else.  At the end of the season, all of the pots are moved into a shed.  I've built special stands to keep the large pots off of the ground.  Makes it easier to 'weed-wack' around them and jog them around to maintain drip lines, do air-layers, prune and such.  Fig trees in any pots that do not have a stand grow roots into the ground every season.  I just pull them up before storing them and they're fine.  Everything gets root pruned every 2 or 3 years, sometimes less.  Move the larger pots around with a 'potlifter'.  Worth it's weight in gold.  Easier with 2 people but I use it by myself, too.

 

There are many reasons that determine the size of your pot, the plant itself can and should adjust to your need. The ideal would be in ground on a beautiful farm in Puglia Italy, ideal but not possible. Fifty years ago I use to cut 50 gallon wine barrels in half, not heavy, easy to moove( for a21 year old) and lot of them around for free. As I got older the pots got smaller, the figs trees don't seem to mind. I now keep my trees in 7 gl pots with lightweight soil, the good thing is I have more time,more varietals and more pots( good thing they are small). Last year I repotted a 25 gl down to10 gallon for my daughter ( originally from Italy), my grandkids harvested about 70 figs from it.

I certainly don't have any video, lol.  It's not hard to imagine.  I casually try to lift the pot with one hand.  If it won't come up I find where the roots are and either pull up or tilt the pot away from the root(s).  They either come up, break or are stronger than I am.  Most either come up or break.  If the roots come up I put the pot away with the roots hanging out so they can grow again next season.  If I can't get the pot up I slice the biggest root with a box knife and try again.  A few are woody but I haven't found any yet that the box knife couldn't handle.  If I can't lift it then it's lather, rinse, repeat.  I cut the next biggest root, try again and so on.  Since there are a maximum of 5 holes and there is only 1 root per hole it can't take forever.  If I haven't answered your question then let me know.

The 7 gal pots are 12-13" tall and the 20 - 40 gal pots are 18" tall

To move my bigger pots I use a hand truck with an extra long tang.  A wet 40 gal pot moves easily that way.  I use this one and it's worked fine but there are sturdier ones out there.

http://www.northerntool.com/shop/tools/product_200586484_200586484

I preffer to stick to below 10 gallons. More than that and ill need two drippers for each tree.

Size does matter, and the bigger the better.  Try for a minimum of 15 gallons.  You can also convert an 18-gallon, storage tub (sold at Home Depot) into a decent, fig tree container.  Just drill/melt drainage holes around the bottom sides of the container - so that if roots grow out of these drainage holes, the roots can be spade trimmed when storage time rolls around.  If you put drainage holes in the bottom of the containers, you will not be able to spade-trim roots if the roots should grow into the soil.  Mulch heavily to prevent the growing medium from overheating if trees are grown in full sun exposures....which is best for the tree.  Avoid shade.

HERNIA WARNING !  These containers are heavy.  Use a dolly to move your trees around.

These container/storage tubs cost less than $10.00 and they last 4-5 years.  Replace if needed, when you do the mandatory, root-pruning.  Use a quick-draining mix, and load it with plenty of pine bark nuggets - to save weight.  All nutrition comes from supplemental feeding when you water the tree(s).  I use Espoma Iron-Tone, with dilute, 24-8-16 Miracle-Grow "tonics" (1 tbs/5-gallons) with each watering.  Stop ferts by mid-July, and make sure that the new growth hardens off properly, and protect your trees before cold, freezing weather starts. 

Good luck and happy growing.

Frank

As an old fart I decided most of my few trees were going up to sip type wal-mart pots when I repot this spring . 
They are  about 10 or 12 gal .. anything bigger will not be manageable .As they go in the cool basement for winter .

I will just have to root prune every few yrs .

I have been using fabric pots as well.  I like 'em.  Easier to move with the handles.  Great drainage.  Into third season without them coming apart.  Doing 1 quart --> 1 gallon -->  7 gallon --> 20 gallon graduations for the faster growing fig cuttings currently; 1 to 5 to 10 to 20 gallon for the slower ones.  So far, so good.