Domain: or get something longer for $9. E.g., authyapp.com or getauthy.com
Design: http://themeforest.net/ $15-$25, or DIY (depends on how much design matters for your niche--for Pinboard it clearly doesn't)
Video: You don't need no stinkin' video
Hosting: Maciej is recommending http://prgmr.com/xen/ for $32 (but can you actually run a modern app stack on just 128 MB of RAM?) Anyway, even if you need a gig, that's $100.
C/Golang/Lua/Vala/SBCL/Ada/Pascal and an embedded datastore like bdb!
Why in my day we used to write little apps with C and bdb that used less than 5mb of memory! (Sqlite didn't exist yo.)
Heck these days you get (ENTERPRISE) choices like LevelDB and BangDB.
Edit: Who wants to have a datastore-backed c10k contest on prgmr machines? :D
I was trying to be lean.
Ruby users are quiche eaters anyway. Real programmers write their code in Fortran and persist to a hierarchical database on a mainframe.
You think real programmers need needles.
In my experience on Rackspace cloud 512MB instances are the minimum you should use for tiny production apps. Also, they're discontinuing 256MB on OpenStack - these were suitable for single-user dev instances, but I got more swap and or locking if ever attempting to run publicly with traffic. 512MB instances solved that issue.
Edit: Also I should mention I'm assuming you mean wanting to run a fairly standard stack, not a stack targeted toward tiny memory footprint.
Depends on what you call modern.
I think fitting nginx / embedded lua / redis should be possible (considering your dataset fits into 32 megs).
Actually, that sounds like an interesting challenge!