The resilience of block fading wireless orthogonal frequency division multiple access (OFDMA) networks to passive eavesdroppers is investigated. The network secrecy capacity is evaluated in scenarios involving a base station and several terminals, some of which constitute passive eavesdroppers. Assuming a block fading Rayleigh channel, the probability of a secrecy outage during a transmission frame is evaluated with respect to a target secrecy rate τ in the following cases: (i) in the absence of any cooperation between the network nodes, and, (ii) when the full multi-user diversity is exploited both by the legitimate users as well as by the eavesdroppers. Remarkably, it is demonstrated that in a network of as few as 12 legitimate users and a single eavesdropper it is possible to transmit 1 bit/sec/Hz with a probability of secrecy outage less than 1%. Furthermore, the delay constrained secrecy capacity of this network is evaluated when the full channel state information (CSI) is available both at the base station and at all receiving nodes. A secure waterfilling scheme is discussed, satisfying a short-term power constraint.