So, how do you suppose people are to purchase houses? Should the houses just be 'given' to people?
As I said above your explicit question seems to be masking a much larger implicit one: what is the nature of 'ownership' or even the 'economy' in an anarchist society? I will try to answer both but by detouring through the larger question and then returning to your specific inquiry.
The answer to the economic structure question isn't entirely clear as anarchists themselves are divided on how an anarchist community would be structured in the specifics (though this is not necessarily a bad thing).
That being said, at its core an anarchist society seeks to do the following (again from an AFAQ):
An anarchist society will try to integrate the social and economic, embedding the latter in the former in order to stop any harmful externalities associated economic activity being passed onto society. As Karl Polanyi argued, capitalism "means no less than the running of society as an adjunct to the market. Instead of the economy being being embedded in social relations, social relations are embedded in the economic system." [The Great Transformation, p. 57] Given the negative effects of such an arrangement, little wonder that anarchism seeks to reverse it.
As I have brought up before, anarchism favors workers owning the means of production in a decentralized and democratic fashion. However as Ender Rises brought up before, simply altering the interior structure of businesses may not necessarily result in conditions any different from the negative social aspects of the current capitalist mode of production. Consequently, this type of anarchist economics could be considered a more 'right' perspective, officially titled 'individualist anarchism', that advocates competition as a means to equality (but this is still not a capitalist system, it should be stressed), though it is not without its problems:
Worker co-operatives of this type do have the virtue of preventing the exploitation and oppression of labour by capital, since workers are not hired for wages but, in effect, become partners in the firm. This means that the workers control both the product of their labour (so that the value-added that they produce is not appropriated by a privileged elite) and the work process itself (and so they no longer sell their liberty to others). However, this does not mean that all forms of economic domination and exploitation would be eliminated if worker ownership were confined merely to individual firms. In fact, most social anarchists believe this type of system would degenerate into a kind of "petit-bourgeois co-operativism" in which worker-owned firms would act as collective "capitalists" and compete against each other in the market as ferociously as the real capitalists used to. This would also lead to a situation where market forces ensured that the workers involved made irrational decisions (from both a social and individual point of view) in order to survive in the market.
To combat the assumed negative effects that such a structure would necessarily result in, social anarchists argue in favor of a more deeply integrated cooperative system that privileges society as a whole and not individual syndicates competing within a specific industry (this could include anything from construction to financial industries where your mortgage question is derived from).
Social anarchists' proposed solution is society-wide ownership of the major means of production and distribution, based on the anarchist principle of voluntary federation, with confederal bodies or co-ordinating councils at two levels: first, between all firms in a particular industry; and second, between all industries, agricultural syndicates, and people's financial institutions throughout the society. As Berkman put it:
"Actual use will be considered the only title [in communist anarchism] -- not to ownership but to possession. The organisation of the coal miners, for example, will be in charge of the coal mines, not as owners but as the operating agency. Similarly will the railroad brotherhoods run the railroads, and so on. Collective possession, co-operatively managed in the interests of the community, will take the place of personal ownership privately conducted for profit." [ABC of Anarchism, p. 69]
Moving yet another step beyond this the so-called 'communist-anarchist' perspective which would seek to have all people, regardless of industry affiliation or not, to have a say in the economic affairs that would affect their social lives. On the other hand, any economic decisions which were deemed largely irrelevant to a larger social community they would not have a say in.
n communist-anarchism, workers make the day-to-day decisions concerning their work and workplaces, while the social criteria behind these decisions are made by everyone.
In this type of economic system, workers' assemblies and councils would be the focal point, formulating policies for their individual workplaces and deliberating on industry-wide or economy-wide issues through general meetings of the whole workforce in which everyone would participate in decision making. Voting in the councils would be direct, whereas in larger confederal bodies, voting would be carried out by temporary, unpaid, mandated, and instantly recallable delegates, who would resume their status as ordinary workers as soon as their mandate had been carried out.
"Mandated" here means that the delegates from workers' assemblies and councils to meetings of higher confederal bodies would be instructed, at every level of confederation, by the workers who elected them on how to deal with any issue. The delegates would be given imperative mandates (binding instructions) that committed them to a framework of policies within which they would have to act, and they could be recalled and their decisions revoked at any time for failing to carry out the mandates they were given (this support for mandated delegates has existed in anarchist theory since at least 1848, when Proudhon argued that it was "a consequence of universal suffrage" to ensure that "the people . . . do not . . . abjure their sovereignty." [No Gods, No Masters, vol. 1, p. 63]). Because of this right of mandating and recalling their delegates, workers' councils would be the source of and final authority over policy for all higher levels of confederal co-ordination of the economy.
Now, to return to your more specific question it is difficult to answer because you have clearly framed it to desire an answer for how we can transition into an anarchist society from a neoliberalist society (i.e. 'would we just give them houses?'), which obviously is a whole other topic that I would rather not delve into at this point in time. Though if you are interested you can check out some of the ideas of how a transition may work (http://www.infoshop.org/faq/secI2.html).
Now the simple answer to your question is that in an anarchist society you can only possess what you use for yourself. As has been argued by Alexander Berkman, anarchism
abolishes private ownership of the means of production and distribution, and with it goes capitalistic business. Personal possession remains only in the things you use [What is Anarchism?, p. 217]
Thus, it can be assumed that if you are in need of a house for yourself the society in which you dwell should be able to provide one to you that is suitable to your needs. And of course, this is all contingent upon the society as a whole, including the individual in need, reaching a consensus or super-majority decision upon the matter and not some institution simply called 'the society' doing it on behalf of citizens.