Just chiming in with my opinion here: I had a similar hosting issue in the past and haven't been able to find a satisfactory "free" solution.
I first started a "private" BLAST server for our offsite collaborators with about 5GB of database sequences. Our university was fine hosting at first, but when bandwidth became an issue, we were asked to migrate the site and database. At the time we were not prepared to buy a server. I think most universities are fine with hosting sites to a certain degree, but when they become cumbersome or require more "in-house" tech support then they prefer to, they ask researchers to acquire their own hosting. I'm not discouraging you from a university server, but I have found our university moved from being supportive to uncool very quickly. My university required that the main person responsible for handling the site be affiliated with the university.
My next step was to investigate free hosting: I found some sites, but no one would host a database other than a small MySQL one (enough for a blog and no capabilities to host any other database managing) and all of them required ads to be placed on the site.
I then paid a small amount (300 US dollars a year) for a server with "unlimited" storage, but when our database grew I was kindly sent an email: "You've exceeded your storage limit; we understand we say this is "unlimited" but if everyone did this we would have no room to host anything. You'll have to pay more to host here."
We eventually bought a server, but we justified it with other capabilities that we needed for our lab.
The AWS Free Usage Tier that Jeroen mentions is excellent, but there's a twelve month free period to get you up to speed with hosting, but then you have to pay. I think it's only an option for people who are just signing up for AWS. If you're comfortable hosting there to get started before eventually paying or moving to a new location, I would suggest giving it a try. If you're looking for a proof-of-concept website for future funding that you can acquire in a year, this might be the way to go.