Is zakat obligatory on money saved to pay off a loan for the purchase of an apartment, given that only a small portion of the loan can be repaid and there is nothing else that can be sold to settle it?

It is permissible to offset the remaining debt with the savings that have completed a full year (hawl). If the remaining money is less than the nisab (definition), there is no zakat (definition) due on it. If the nisab or more remains, zakat becomes obligatory on the remaining amount. It is important to note the prohibition of borrowing with riba (usury/interest), as it is among the gravest of sins.