Hi Savannah - and welcome. Just a quick thought. The replies to your post all argue that because the barrel has not been maintained to hold liquid over the years it is now almost certainly less than fit for that job (barrels deform, staves "leak" and all kinds of bacteria and strains of yeast make their home in the wood once used to store wine. But your initial post suggested that you wanted to learn more about using barrels. What is it that you want to learn more about? The answer to that question may take you down a slightly different path. I myself don't use barrels but barrels can provide a number of benefits to a wine; the barrel itself allows for micro-oxidation (air does get into the barrel), provides additional flavors from the oak - often thought of as vanilla-like; and will provide flavors from previous batches that were stored in the barrel (which is why distillers, with some exceptions, like wine barrels). But the thing is that the "flavors" that are extracted from the wood itself can only be extracted so many times before they are totally depleted. If a barrel was used to age wine three or four times that barrel no longer will provide any real oak flavor.
But here's the thing: You say that the barrel your mother received was 60 gallons. Sixty gallons for most home wine makers is huge. Most home wine makers make perhaps 5 or 6 gallon batches, sometimes 3 gallon batches and if they don't make wine from kits they may make single gallon batches. And often because of the relatively small amounts of wine home wine makers make rather than put their wine in barrels they stand that idea on its head and put barrels inside their wine. By that I mean they buy toasted oak cubes or spirals of oak or staves and stick etc and they add these bits of wood to the wine in their glass or plastic carboys. Barrels can cost a lot of money to maintain - and a great deal of time and effort - but oak cubes cost pennies.