As a privately-held company, your only options are public or proprietary databases to access "non-confidential" information. Sites like Crunchbase often include financial information specific to funding rounds. Similarly, corporations (even private corps) often file forms with the SEC that are available for search through EDGAR.
Looking at their management team, they appear quite seasoned in the sense that they may have access to capital through other means than private equity funding when compared to a typical startup where the founders and management team are younger and often first-time entrepreneurs.
Depending on your reason for finding out specific financial info, you can always contact the company to conduct an interview. Otherwise, as a private entity, they're not required to disclose any specific information. I'm sure you know this but it's worth mentioning.
Relative to their cost model, it's hard to estimate their profitability today or in the future. Without any reliable data source, running a model based on estimates results in a model that's, well, an estimate. With that said, you could always model the infrastructure costs (storage, bandwidth, etc) to find out what their costs are. For example, a bit of hacking public info by way of DNS queries, viewing-source, etc might suggest they leverage a certain cloud hosting provider with a CDN and certain storage network. From that point, you can extrapolate the costs based on different usage levels.
Taking a few steps back, if you're interested in figuring out how/why they're in business, I'd recommend resources such as Techcrunch and Betabeat to study business models and the dark art of valuation. Companies don't need to be cash flow positive (i.e. profitable) to continue operations. As a recent example, take a look at Instagram as well as other acquisitions completed by Facebook, Twitter, and other social media companies.