The concept of a tradable asset is closely related to the principle of (no-)arbitrage. Much of quant finance is about the connection between the price of a derivative and the price of its underlying. The fundamental reason that there is a connection at all, is the possibility to set up self-financing trading strategies in the underlying(s) which replicate the pay-out of certain derivatives. If you look at the definition of self-financing strategies you will notice that you need to be able to buy and sell the underlying at any time, instantaneously in unlimited quantity, without any transaction costs. So this is a tradable asset within the mathematical theory.
Of course this is mathematical fiction and imposes an important restriction on the applicability of mathematical finance to the real world. But from my (limited) experience more realistic assumptions quickly become messy and really difficult.
In my opinion being tradable in the real world is not black and white but a gradual thing. So some examples in descending order of tradability: Bank deposits, S&P futures, derivatives on the ABX index , your house(or car or bicycle), the Sistine Chapel.