He seems to live quite well, but unlike other hobbits he doesn't look like a farmer. Was he a landowner or something?
Bilbo was just born into a wealthy family; see Letter 25:
The full list of their wealthier families is: Baggins, Boffin, Bolger, Bracegirdle, Brandybuck, Burrowes, Chubb, Grubb, Hornblower, Proudfoot, Sackville, and Took.
Much of his pre-adventure wealth was therefore inherited; his parents had built his Hobbit-hole, and they came from two on the list of wealthy families: Baggins and Took.
He's very much a stereotypical idle upper-middle-class layabout bachelor, not quite a Bertie Wooster but not too far off one either.
Tom Shippey's Road to Middle Earth includes a discussion of his character in this context ("The Bourgeois Burglar") which you may find further illuminating.
Bilbo was a wealthy landowner like Tolkien was familiar with in British society society and from literature.
Apparently the Shire as originally settled in Third Age 1601 by powerful hobbit leaders who led groups consisting of their relatives and followers and dependents who settled in various regions and claimed the vacant and long abandoned lands in those regions.
By Third Age 3001 there were still many traces of that old clan like social structure.
Thus most of the wealthy families were headed by chiefs who had some degree of authority over other family members, even distant cousins, and their tenants and dependents and poorer neighbors.
In one of his notes or essays Tolkien wrote that the ambition of the Sackville-Bagginses, chiefs of the Sackville clan, was to become chiefs of the Baggins clan also and thus be double chiefs.
In the days of Bilbo and Frodo members of old aristocratic families no doubt owned many farms and were wealthy from rents from tenant farmers, though no doubt there were also many small farmers who owned their farmland.
The Shire was divided into four quarters, the Farthings already referred to, North, South , East, and West; and these again each into a number of folklands, which still bore the names of some of the old leading families, although by the the time of this history these names were no longer found only in their proper folklands. Nearly all Tooks still lived in the Tookland, but that was not true of many other families, such as the Bagginses or the Boffins. Outside the Farthings were the East and West Marches, the Buckland and the Westmarch added to the Shire in S.R. 1462.
Appendix B The Tales of the Years, says that the Westmarch was added to the Shire in S.R. 1452 (T.A. 3052 or F.A. 30) by the gift of King Elessar.
Since the hobbits had very little government in the Shire, it is very hard to tell if the Buckland was part of the Shire - and merely another folkland - or totally independent of it.
The Shire was part of the Kingdom of Arthedain from the Shire's foundation in T.A. 1601 until the kingdom was destroyed in T.A. 1974 (S.R. 374).
But in that war the the North Kingdom ended: and then the Hobbits took the land for their own, and they chose from their own chiefs a Thain to hold the authority of the king that was gone.
Pippin (Pergrin Took) and Merry (Meridadoc Brandbuck) eventually became Thain of the Shire and Master of Buckland respectively, and thus not merely chiefs of families and folklands, but more like kings or dukes of small kingdoms or duchies. After all the Thain had the authority of the former king - or as much of it as the Hobbits wanted to grant to the Thain. Although with the lack of strong government authority in the Shire, and the renewed overlordship of the King of Arnor and Gondor, they had more high status than political power.
In the Hobbit at the "unexpected party" Bilbo asks Thorin about terms and conditions, using the voice he used when people tried to borrow money from him, if I remember correctly. So Bilbo and other wealthy Hobbits may have had income from lending money to other hobbits.
At the time of LOTR Lotho Sackville-Baggins, a clan chief and wealthy landowner, also made a lot of money exporting pipweed and other goods to far off Insengard.
And of course some hobbits, including wealthy landowners like Bilbo, also made money selling food and other goods to Dwarves passing through on the road. No doubt there were also business deals or even partnerships between Dwarves and Hobbits to import and export goods to and from the Shire.
Considering the exotic and foreign plants and objects (like Dwarf made mechanical clocks) owned by Bilbo, he must have been quite aware of imports into the shire, and may have had a financial share in some importing activities.