This is a partial answer, in the sense that it addresses the case for state-sponsored house building, rather than the question of how that should be financed. In general, any means by which the state might finance spending is likely to introduce some kind of distortion into the economy. The question then faced is whether that spending generates benefits sufficient to offset the cost of the distortion.
I agree with the general sentiment of others that if there is a well-functioning, uninhibited market for housing then any intervention in that market (such as an artificial increase in supply) is likely to reduce overall efficiency and, in particular, lead to the misallocation of resources within the economy more broadly. @suriv outlined the intuition for this.
However, if there are distortions that prevent the housing market from functioning efficiently then a state-sponsored increase in housing supply can be efficiency enhancing. This is a simple application of the more general "Theory of the Second Best", which holds that if there is pre-existing distortion in the economy then introducing another distortion can sometimes make things better, not worse.
There are various reasons why the housing market might not function efficiently. Here are a couple:
Planning laws are sufficiently restrictive so as to reduce the market's ability to supply housing. For example, if planning permission is hard to obtain then developers who have planning permission have a kind of monopoly power over the ability to build houses and are likely to ration supply in order to increase profits. N.B If this is really the problem, then reform of the planning system is likely to be more efficient than a direct housing market intervention. But such reform might also be politically problematic.
Property builders are extremely vulnerable to economic downturns because housing demand is pro-cyclical. They are therefore unwilling to commit to risky, large-scale construction programmes that have long time-horizons.
Speculation on rising property prices induces people to hold undeveloped land or un(der)occupied properties in the hope of selling later when the price is higher.
All of these factors seem to be important considerations in the UK market (which is, I guess, the example that motivated your question). Since we are talking about state-sponsored house building, here is a graph that shows how much of the slack private-sector housebuilders in the UK took-up when the state withdrew from house building in the 1980s:
The answer is "basically none".
But does that mean that the market is now failing to supply enough housing, or just that there was previously an inefficient excess of supply that the market has now corrected? A figure that, I think, is broadly indicative of the fact that the UK housing marketing is currently not functioning well comes from this UK government report. It states that the UK is currently building houses at a rate of 135,000/year, but creating new households at a rate of 240,000/year. The 240,000 figure does not include either backlog from undersupply in past years, or demand for housing from foreign speculators (the FT reports that 75% of new homes in London are bought by foreigners, many of whom are speculators or will otherwise not occupy the property full-time). The UK has never in its history managed to build 240,000 homes per year except at times when the state accounted for close to half of all house building.
As somewhat of an aside, the effects of this long-term failure to meet demand upon the price of UK housing has been significant. Here is a comparison of various national house price indices from The Economist:
If one believes that such price growth is the consequence of a distortion in the market then it has important knock-on consequences for other aspects of economic efficiency: it encourages speculation and high levels of indebtedness, distorts investment decisions, and induces internationally mobile workers to leave the UK. As regional price disparities are magnified by rapid price growth, internal migration is also potentially distorted. If we care about equality then this kind of price growth also implies a growing wealth gap between property owners and non-owners.