I think the question we should ask ourselves is not how to eliminate inequality, but what baseline level of services and opportunities we want to give people at any stage in their life, regardless of their competitiveness and productivity, including when they're otherwise poor and a net drain on society.
In other words, is stable housing a basic right or is it a luxury? If it's a basic right (and/or a net positive if it gives people the chance to bootstrap themselves into higher earnings), then we should strive to increase the reach and quality of tax-funded government housing programs - together with other baseline programs, such as public education, social services, etc. Or if you're a small-government person, substitute "tax-funded" with "charitable" and make it dependent on individuals' goodwills.
If it's a luxury then the status quo seems acceptable - markets need to be able to pick their customers, and baseline services are losing products unless the "exploitative" pricing makes up for it.
Abolishing inequality isn't desirable at all, but the range of inequality doesn't have to go from 0 to billionaire. We can provide a baseline that still leaves enough incentive for people to work their way up but is decent enough to live with dignity, greater than 0. Soviety provides some of that today. Discuss where to draw that line and who, taxpayer or otherwise, can provide it. A functioning society is never completely equal or completely without minimum support, always somewhere in between.