Yes, you can own a home and qualify for disability, but the exact rules differ depending on which disability program you're applying for. The Social Security Administration (SSA) administers two programs: SSD (or SSDI, Social Security disability insurance) and SSI (Supplemental Security Income).
The SSDI program, paid for out of workers' tax contributions, is not the least bit concerned with the subject of assets, called resources by the SSA. So, you can file for SSDI whether you own a single home or multiple houses or vacation homes or rental properties. SSDI is also not concerned with other types of assets such as multiple vehicles or investment accounts, and so on. In short, assets do not affect eligibility for Social Security disability insurance.
SSI is a different matter. SSI is a need-based program for individuals with low income and low resources. Individuals can apply for SSI if they aren't insured for SSDI or are insured for SSDI but are only eligible to receive a fairly small SSDI monthly benefit (in which case a person may potentially be eligible to receive both SSD and SSI -- this is known as a concurrent disability claim).
How do assets affect a person's eligibility to receive SSI disability benefits? Currently, the cap on assets for SSI is set at $2,000 (or $3,000 for a couple). But not all assets are counted toward the cap on assets (the SSI resource limit). The house you live in and the car you use for your basic transportation are usually not counted. As long as you live in your house (or intend to return), your house should be excluded, even if you share ownership with someone else. For more information, see our article on when Social Security will exclude your house from being counted as an asset for SSI.