It is probably possible, just very hard and there is a very high chance you will lose very fast. I would suggest taking at least one other person who is close to level 20 or at level 20 with you. Also you should be close to that point yourself.
Let's break it down. Enemies, when spawned will directly move towards your location or towards a vantage point to fire from. This happens in every party situation - from one to four players. What 4 players does is to divide the enemy attention so that it is more manageable. This also allows for flanking with one part member as distraction or sentinel role (FFXIII) reference. When going solo, all enemies will DIRECTLY TARGET you, and you will not be able to flank without cloak. In addition, the amount of enemies in multiplayer is balanced by the ability to revive other players without medi gel and the fact that all players are revived and restored to full health after each round. Coupled with the insta kill enemies (Phantom, Atlas, Banshee) this makes going solo extremely hard. If you REALLY REALLY wanted to try gold on solo, it's almost required to be an Infiltrator. This will take an extremely long time, but with a powerful sniper rifle, you can slowly thin out ranks and take out foes. After two teamates and I died, this is what some guy did against reapers on SILVER. And he still got owned by a banshee. I think the only possible way of doing gold on solo is with Quarian Infiltrator against geth - invisibility and sabotage will help even odds. But multiplayer was DESIGNED to be for multiple people (duh). Unless you're MLG level good, you're probably screwed round 2.
-It is certainly possible. I myself have gotten, on average, to wave six on gold. The key is, as the above poster posted, to cloak. I am willing to wager that it is nigh impossible as anything other than an infiltrator. While Quarian infiltrator does sound like a good choice, Salarian is better the proximity mine is a twofold force multiplier. When you cloak and then simultaneously fire a round from the Widow with a proximity mine, you will take out a quarter of an Atlas' armor plus the splash damage from the mine will do massive damage to surrounding units. I would only suggest using this against Cerberus and the Geth, as banshees are going to be a bitch, no matter what.
-- Michael Gambel says no group has managed to defeat the Reapers on Gold, so you probably cannot solo them. (http://twitter.com/#!/GambleMike/status/179336226116079617)
There are people who soloed gold against cerberus. So yes, it's possible. And they always use salarian infiltrator equipped with a widow.
I must say, I find this immeasurably improbable, on the rounds where you just have to survive the waves, sure, I've cleared a few of those solo on an infiltrator, but when the ones with uploads and activations/deactivations come across, you're utterly screwed, As an infiltrator, you can handle the activations somewhat painlessly, but the uploads are nigh impossible. touny