I'm playing as a level 5 warrior as the moment, focusing on Block, Blade, and Heavy Armor. I decided to do the Fighter's Guild questline to level up a bit, then move on to the rest of the Arena questline, and then finally the main questline. I'm only trying to use realistic things, such as Steel Armor and a Steel Longsword.

But you can't.

I'm currently doing Drunk and Disorderly, and I have made my way to Leyawiin, spoken to the 3 guild mates, and been to Margarte's. She wants 5 ectoplasm, and I read that the Temple of Zenithar across the street has 5 ethereal beings in it who drop ectoplasm in the Undercroft. I am in the Undercroft, but I can't kill them cause my weapon does no damage.

Do I need an enchanted weapon? If so, where can I find one at easily?

You need either an enchanted weapon, or one of silver, or daedric material. You can check the shops for magic weapons, or failing that go dungeon diving. Bandit dungeons are a good place to start. Rockmilk Cave is nearby, and it will have lots of gear.

You can also kill the ghosts with fire spell which you have by default.

I used the fire spell, but nothing seemed to be happening. Perhaps I should lower the difficulty just this once to get past this roadblock.

I do not have a silver or daedric weapon, and I doubt I have enough money for an enchanted weapon. Perhaps I can find a cheap iron dagger with a basic enchantment somewhere?