This is a stylized or literary way of saying problems one is unable to repair or resolve, because one's ability to do so is insufficient. This is a fairly formal construction, and less likely to appear in colloquial speech.
Beyond X's Y is an idiom meaning outside, past, or greater than the scope of Y, as Y relates to X. Examples:
- Beyond one's ken - outside of one's knowledge and experience
- Beyond my means - it would take more capacity, resources, or abilities than I possess or have access to
- Beyond the pale - past or overreaching the appropriate boundary, in an egregious fashion; one is understood to mean everyone or the topic in general
In this specific case, fixing is a gerund, and as such, an appropriate target for the preposition beyond. Because this is an idiom, it's understood that its meaning here is ability to repair, and not literal act of repairing.