Is it (practically) possible for a large building to be a Faraday cage?

In general the answer is "yes it is possible" - but in your case the answer is "that is not a Faraday cage".

Radio waves are (partially) reflected by any discontinuity in dielectric constant of the medium they propagate through. The ones that propagate (through walls etc) will also experience attenuation.

A faraday cage is a continuous conducting structure with no openings that are "large compared to the wavelengths of interest". Your building has windows that are much larger than that. The wavelength of a cell phone signal (typical frequency 1800 or 1900 MHz so around 15 cm) is small compared to windows and signal would penetrate - meaning that it is not a Faraday cage.

On the other hand walls do provide significant attenuation depending on the material - and waves that have to diffract through the window would also be much weaker when they got to you. If the gym was sufficiently far from the nearest cell tower it is easy to get a "dead spot" in reception.

For reference, according to this link a concrete wall provides 10 to 15 dB of attenuation - which may be enough to drop the signal from "OK" to "not OK", depending on the signal strength outside.

Just to add to what Floris has said. It is frequent (in the UK) that institutional settings would have toughened glass in windows, particularly in bathrooms, gyms etc. that would have the form of a wire mesh (of order 1cm grid) embedded in the glass. That would do a particularly good job of blocking phone signals that would otherwise penetrate the glass.