Can you just be a merchant?

I was wondering if it's possible to just level the crafting/gathering classes without leveling a regular class?

I have an idea in my head just to level a merchant type character.

Also, what are the most useful crafting skills to have right now?

You wont be able to progress through the main story with a craft or gather class, but once you hit around lvl 10 or 15 they allow you to change classes so if you want to just do just crafting or gathering without the main story you can do that.

You will want to get through the main story far enough to unlock things like retainers, airship travel and chocobos, ideally, as you'll find them invaluable for your trading (unless it's purely RP). Fortunately this doesn't take that long and is quite fun to do. If the battle classes still haven't won you over by then, you can focus on your crafting and it should keep you busy for a while. My friend hit level 50 as a miner long before he finished working on his bard.

Short answer, No.

Long Answer:
Although you technically could level a crafter/gatherer classes without doing any of the dungeons or story quests, you will eventually run in to a wall where you can't actually obtain the materials you need for your crafts. Fortunately it is easy and you can get a single class up to 60 simply by doing the main quests and side story quests. Most of the side stories just expand on the lore, so they're not exactly dull. The story itself is mediocre for ARR but it becomes very good in Heavensward. Even the voice actors changed and it's a lot easier to listen to; although you still have to put up with the archaic english.

The limit on combat is not only for mats and gathering nodes in Heavensward (which requires a lv.50 combat class to unlock the story), but also for air travel (to get to specific nodes on islands and is gated behind sidestory quests), treasure maps, and other methods to get rarer materials. The new airship expeditions in the upcoming patch could work with only lv.60 gatherers, but you might need a FC or party with some lv.60 combat classes helping to keep you safe.

You can have your retainers collect these materials (if you're lucky), or simply buy them off the marketboards, but it will cost you around half a million gil for each completed item at Lv.60. It might be worth it simply to game the marketboard though.

P.S. You will want to level all your crafters to 15 first and then to 50, but the most useful ones are:
ALC: Tricks of the Trade and Comfort Zone
ARM: Piece by Piece
BSM: Ingenuity
CRP: Rumination, Byregots Blessing
CUL: Hasty Touch, Steady Hand II
LTW: Wastenot II
GSM: Manipulation, Flawless Synthesis, Innovation
WVR: Careful Synthesis II

If you're starting out as a crafter, just get Tricks of the Trade, Rumination, Hasty Touch, and Manipulation; all obtained at level 15. You can HQ reliably from that point on and level up more efficiently with your leves.

Also, unlock the Ixal beast tribe quests by completing the lv.41 story quest, In Pursuit of the Past, for some massive xp for crafters and crafter-oriented daily quests.

Well, short version... there's a minimum (a very easy minimum) you need to do on a battle class, and then you can (and some people do) spend all your time on crafters and gatherers. Or if you mean merchant in the purest sense... you'll need retainers so you can sell stuff, then just spent all day clicking the MB and you can get by just fine.

You can level up a single combat class to 60 in order to get past the MSQ and unlock Idyllshire (at lv. 58). Then, you're free to spend all your gaming time on crafting and gathering if you wish to.

The first few levels you have to do with a combat class. After that it depends on whether you want to be doing the top level endgame crafting/gathering or not.

Combat class requirements are:
Level 10: You have to get this far and complete your level 10 class quest before you can even add additional classes, so you cannot even begin crafting before this point.
Level 17: You have to get up through the Scions of the Seventh Dawn quest to unlock access to retainers. You cannot sell stuff on the MB or have much in the way of item storage without this.
(Level 20: You have to join a Grand Company to get a chocobo. Not needed for crafting, but certainly helps make gathering easier.)

For the next 30 levels or so, progressing a combat class is helpful, but not strictly necessary. You need a combat class to unlock levequest locations at each level range and need to get it up to somewhere in the early 40's to unlock the Ixal beast tribe quests. Both of these provide ways to level your crafting classes faster. But if you're not in a hurry and are willing to level them the slow way by just gathering and crafting stuff, you can do that without continuing to level your combat class.

If you want to get into the top-end crafting in Heavensward, then you'll hit the issue Anova brought up that you'll reach a point where you can't reach high end materials without unlocking the new regions, and doing that requires taking a combat job all the way up into the late 50's and through the whole storyline to that point.

However, a casual player with lots of time but no pressing desire to make the top-end items requiring HW materials could do quite well for themself just continuing to make and sell the level 1-50 items made from materials available in ARR regions. There's some stuff they couldn't make, but plenty else that they could. For this approach, that level 17 unlock of retainers is all that's really needed from your combat class.