As a first time writer, most publishers insist that you approach via a literary agent. However, all the literary agents I have contacted claim to be too busy to take on new clients, as they receive hundreds or thousands of submissions every year.
I used the Writers & Artists Yearbook to get a list of likely agents, then checked all their websites and eliminated ones that didn't seem suitable. For the rest, I noted their precise submission requirements (lots of variation, but I did exactly what they wanted). I submitted to the agents left on the list, exactly as they specified, and so far of the responses I have (20+) nobody has said they didn't like it, in fact several have said that it sounds fascinating. However, they all claim to be too busy to take on new clients and wish me luck.
The impression I'm getting is that getting an agent is an impossible requirement if agents aren't accepting any new clients due to their existing workload.
If I genuinely believe in my project, and everyone who has heard about it says it'll be great (not just friends and family, but random people too), where can I go from here?