You download this version from the main Oracle "Latest Downloads" page.
You must own a valid Oracle Database license to download 10.2.0.5. There is no free trial for this legacy software without a support contract. Oracle Database 10g Release 2 10.2.0.5 Download NEW
If you must download it, log into → Patches & Updates → Search by Patch ID (8833280 for Linux x64). If you do not have a support contract, consider using Oracle Technology Network (OTN) for 10.2.0.1 (the base release) and then use the publicly archived documentation to request the terminal patch via a customer service ticket. You download this version from the main Oracle
Remember: With great legacy power comes great responsibility. Secure your 10.2.0.5 database as if it were made of glass—because without modern patches, it is. Oracle Database 10g Release 2 10.2.0.5 Download NEW