Storing database files on network storage is probably not a good idea with regard to performance; and probably security. Make sure your network is fast enough, and the connection very reliable.
Anyhow, it is feasible.
You seem to be working on Windows. Depending on version and mode of installation, the user running your PostgreSQL service might be different.
First thing to do: check which one it is. Use the "Services" administration tool, find out a service with a name similar to
Postgres, and find out which user is the one that will be running PostgreSQL. In most recent versions and installation modes, it is commmon that this user is "Local System" or "NetworkService". In previous versions, it used to be a Postgres local user (i.e.: not an Active Directory domain user).
At this point, you have two options:
Change the user (and password) to another one that has permissions to access your network share. Create one if you must. This user must not have administrative privileges; and it should have privileges only to be able to handle the database (i.e.: access to the part of the filesystem that it requires, which will be the PostgreSQL data and log directories that you chose while installing and all the tablespaces you define).
Give the existing user
read/write/modify permissions to access the network share. You shouldn't (and you probably just won't be able to) if the user is one of the standard Windows, that would most probably open the whole Windows to lots of security risks.