The Institue of Space Sciences (ICE-CSIC) can be found in the Campus of the Universitat Autònoma de Barcelona, approximately one hour from the city centre. The campus is very well connected by public transport, being at a 10' walk from the Universitat Autònoma station.
From Barcelona city centre
From Plaça Catalunya, catch the train (FGC) bound to Sabadell Parc del Nord (S2), and get off at the Universitat Autònoma station. Plaça Catalunya is very well connected via train and metro to the rest of the city.
From Avinguda Meridiana / Carrer de Felip II (metro L1 / L5, Sagrera) or Avinguda Meridiana (metro L1, Fabra i Puig), you can take the E3 bus towards UAB (Universitat Autònoma de Barcelona) and get off at UAB-Ciències.
Alternatively (though less frequent and less punctual), you can take the R7 train from the Sagrera or Fabra i Puig stations (it runs more frequently from Fabra i Puig). Get off at the terminal station, Cerdanyola Universitat. From there, you can either walk for about 10 minutes or take the L1 bus and get off at UAB-Ciències.
Information about the different public transportation ticket options is available here.
From the J. Tarradellas Barcelona-El Prat Airport
Get on the shuttle bus (A1 / A2) bound to Plaça Catalunya (approx 30'). Cheaper (but slower) alternatives are the metro (L9, change to L1) to Plaça Catalunya or the local train (Rodalies) to Passeig de Gràcia (only available at Terminal 2). At Plaça Catalunya, catch the train (FGC) bound to Sabadell Parc del Nord (S2), and get off at the Universitat Autònoma station. If arriving by train to Passeig de Gràcia, take the same train from the Provença station.

