In the same way that this is correct (though perhaps unusual) English: "That I cannot afford."
However the Dutch word order starting with dat is a common one, whereas your sentence is unusual in English as you mention. Usually word order in Dutch is quite flexible, in most cases there are a number of correct (and commonly used) options. English word order is a lot more strict (and restricted).
Can anyone confirm that ''veroorloven'' is actually pronounce in the way the MAN is saying?
I got a suggestion here for "I cannot allow that" - so can veroorloven also mean allow?
Copying PedroChopi's reply, so it doesn't get separated from this question:
Veroorloven means to allow. The reflexive pronoun is what makes the sentence mean I cannot allow myself (to buy/do) that.
If someone had just said "ik drink vijf bieren elke dag" and you reply with this sentence, are you implying you couldn't afford that many beers? Or that you couldn't allow yourself to do that for some unspecified reason (health/morals/etc)?
All are possible, in fact you could add money to your list of unspecified reasons. :)
Ik begrijp het, bedankt :)
How would you imply that money is specifically the problem? Is there a better word for that than veroorloven?
You could either make it clear by context, e.g. continuing the next sentence talking about: your lack of money, being careful with your money or telling what you do spend your money on.
Or you could say something like: Dat kan ik niet betalen / Dat zou ik niet kunnen betalen
In Spanish, when we say "eso es algo que no puedo permitirme" the implication is that you cannot afford it. If it is something that you cannot do for moral reasons, you would say something like Dat mag ik niet doen...