The normal statute of limitations for burglary is three years. However, that time limit can be extended in certain circumstances, usually involving misconduct such as the defendant hiding to prevent prosecution. If no charges are brought, you can seek to have your arrest record expunged, given that you meet the other requirements for expungement. Do not post any other facts of your case online, wait out the limitations period, and hope that you are not formally charged.
The statute of limitations for burglary is 3 years. If no charges are filed, you may expunge the record of arrest after a year.
As Mr. Crow and Mr. Turnoy have noted, the statute of limitations is 3 years. However, Mr. Crow importantly points out that you can expunge your record of arrest after 1 year which sounds like it might solve your problem with obtaining housing. To determine if you are eligible for expungement look at Oregon Revised Statute 137.225.