See discussion area. As I've said there, "bajera" is a word commonly used in Navarre, and also (as Ion says) in the Basque country, to refer to a room on the ground floor. It's not used, as far as I can tell, in other parts of Spain. Since the person who used the word in this text was apparently from Navarre, that must be the meaning.

The problem is what to call it in English, because it could refer to several different things. It is often a room in a house, which can be a garage, a store room or a utility room. It can also be a ground-floor commercial unit in a street. It can be what we call a lockup in the UK. But we don't know which of these it was in this case. People sometimes seem to have a "bajera" separate from their main residence. It can be very small, but it can be quite large (60 m2 or more). So I think the best thing is to play it safe and just call it a ground floor room. This is a more general word than I would like to use, but I don't think we're in a position to be more specific.

"Basement" actually evokes the right kind of image, since basements in houses are used for some of the same purposes as "bajeras": storage, maybe DIY, playing table tennis, or just hanging out and partying with friends. But a basement is always below ground level and a "bajera" is definitely on the ground floor.

Here are a few references that give an idea of what a "bajera" is or can be. First, some pictures. Ignore the sheets ("sábanas bajeras"):
https://www.google.es/search?num=100&biw=1024&bih=653&q="baj...

More "bajeras" here, some quite large. These are more like commercial premises:
http://www.inmobiliariafuentedelhierro.com/inmuebles.php/tip...

As I said, they're popular with young people:
"Somos un grupo de jóvenes de entre 17 y 18 años que buscamos una bajera para pasar las tardes. Precio máximo 400 euros y que sea de 50 m2 como mínimo"
http://www.milanuncios.com/alquiler-de-locales-comerciales-e...

But they're often rooms in houses. They can be used as garages. Here's one from La Rioja (by 1ª planta they must mean ground floor; a garage on the first floor is not much use!):
"Casa en alquiler y su precio es negociable. 1ª planta bajera para 2 coches mínimo."
http://www.enalquiler.com/alquiler_casa_calahorra/alquilo-ca...

Here's one that's a separate storeroom, like a lockup:
"Contamos un bajera-trastero situada en una calle aledaña donde guardamos bicicletas que también están disponibles"
http://www.intercambiocasas.com/show.php?id=163487

And so on. There are examples that are like converted apartments. The one thing they seem to have in common is that they are on the ground floor.