UPDATE - Looks like this was added as an option after I posted this. Leaving the post here since it explains both versions. Just be aware that now you have the option when you setup a table to play the standard (elimination) or the variant mentioned below.
More info. The game version on BGA appears to be using the variant discussed in the FAQ under "Shipyard" (see below), so if ALL your ships are made obsolete and you are out of credits you get a lease for the cheapest available ship vs. being eliminated from the game.
Q: If I have no Ships due to obsolescence and not enough credits to lease an available Starship, what happens?
A: You are eliminated from the game. If desired, play this variant: a player without a (non-obsolete) Starship
and without the funds to lease one, must turn in all of his or her credits to lease the cheapest Starship
available for purchase. However, a player may not place any retained gear or Pilot on this Starship. Instead,
any retained gear or Pilot continues to be retained until a player later buys/leases a Starship normally.