What could I do to improve LaunchAStartup, a freelance marketplace that competes with Elance, oDesk, et. al?

Looking quickly at your website, I would bounce quickly. Here's why: The jobs posted look like things I'd find on Fiver or other low-quality, "micro-task" service marketplaces. In other words, your landing page doesn't deliver on the brand promise inherent in your name.

I'd take a totally different approach to your site which would include forcing startups interested in your service into a sign-up form where they describe specifically what they are looking for, and keep the landing page focused on any social proof (ideally, happy customers) and describe the various types of services available in your marketplace.

This will allow to better assess the needs of your potential customers and allow you to better direct work to the freelancers registered that you know (or think to) be good.

That's just the tip of the iceberg. If that doesn't sound crazy, happy to provide more detail in a call.

Here's a few thoughts for you.

1) I go to Elance / Odesk and know that I will virtually always find someone to do what I need, no matter what it is.

Programming and design work are more than likely two of the largest areas of these types of sites.

You have 9 people offering programming services and 16 people offering design services. That is simply not enough choice. Considering the big guys have hundreds (if not thousands) of people in these categories. You're not even close to being "10% as good" as your competitors.

Considering the people offering services are essentially your "product", you don't have many at all.

2) When people are searching for people to hire, you should make the job images, along with the job title clickable. To find out more information I currently have to click a little blue link under the image and text description. It's not intuitive design.

Along with the that, once a category is chosen, IE programming you need to be able to narrow the search even more.

It's not a problem now as you don't have many freelancers but eventually, crawling through 20 pages to find 1 Wordpress will not be acceptable.

You say that conversion rates are 'dismal'; are you sure they cannot be optimised further? What level of conversion would you need to make your current business model work for you?
Optimising your conversion rate is all about the landing page. Minimise clutter, clarify your core message and immediately engage the user. A/B testing should help you optimise. Get as much outside feedback as you can and don't stop pushing until the metrics show you're right.