Does the place become impure, and is it permissible to pray in it, after relieving oneself in a drain inside an enclosed space, provided one ensures that the impurity does not spread, performs istinja, and pours water into the drain?

There is no impediment to what you are doing if the place belongs to you or you have been granted permission to relieve yourself there. The entire place is not considered impure, but only the part touched by the impurity. The place where one relieves oneself or the part touched by impurity is purified by pouring water over it until the impurity is completely removed. It is stated in "Al-Mubdi' Sharh al-Muqni'" that impurities on the ground or anything connected to it are purified by drenching them with water, citing the hadith of the Bedouin who urinated in the mosque, and the Prophet commanded that water be poured over his urine.