You can actually complete the quest and get the sword with any class, but you will NOT get the achievment if you can't equip it.
Is combined with Eye of Sulfuras to make the legendary weapon Sulfuras, Hand of Ragnaros.
First of all i want to say that all of the mats for this weapon and the weapon itself are not bind on pickup meaning you can buyall of those from AH. But if you are willing to craft it yourself here's a list of what you'll need:
Sulfuron Ingot - is obtained from a Molten Core boss called Golemagg the Incinerator with ~22% drop chance. So if you are not going to buy these on AH or if you are not farming Molten Core every week on main and alts it can take quiet a few weeks before you can get enough of these.
Dark Iron Bar - is smelted from 8x[Dark Iron Ore] each. Dark Iron Ores are obtained from mining Dark Iron Deposits. Also some mobs and bosses in Blackrock Depths.can drop 1-2 ores each but the chance is quite low so it will be much faster to mine it which can also get you some Bloods of the Mountain.
Blood of the Mountain - can be obtained from giant lava mobs in Molten Core called Molten Destroyer or can be mined from Dark Iron Deposits. But overall it's easier to mine Dark Iron Deposits then farming Molten Destroyers. because the drop chance is quiet low. Anyway getting 10 of these without using AH probably will take some time so be ready for some farming.
Arcanite Bar - is transmuted by alchemists using 1x[Thorium Bar] and 1x[Arcane Crystal] each. Arcane Crystals (as well as Thorium Ore which is smelted into Thorium Bars) are obtained from mining any type of Thorium Vein. Good way to farm both ores and crystals is mining in Silithus / Un'Goro Crater. Transmute masters can get procs makng Arcanite Bars.
Essence of Fire / Fiery Core / Lava Core - are easily obtained from trash mobs and bosses in Molten Core so you should have plenty of those after defeating Ragnaros couple of times.