Is 80 bits of key size considered safe against brute force attacks? I came across the KATAN family of ciphers for small domain input blocks. They cipher arbitrary block lengths: 32, 48 and 64, but their key size is 80 bits only. 

Is 80 bits of key size considered safe against brute force attacks with current state of the art processors?
TL;DR: no longer unconditionally.
As of 16 Feb 2021, the Bitcoin miners hashed at an aggregate rate of $\approx154 \cdot10^{18}H/s$ according to this source, where one hash is two nested SHA-256; that is $\approx2^{93.0}$ SHA-256 per year.
Here is this data redrawn in SHA-256 per year with a $\log_2$ vertical scale, to facilitate comparison with key size in bits. I believe the notable growth in 2010-2011 was the rise of GPUs, while the 2013-2014 period corresponds to an increased use of ASICs. Nowadays, bitcoin mining is mostly performed using highly specialized ASICs. Recent hardware advertises 220 T SHA-256/s at 15 pJ/SHA-256 (3250 W power on wall) for \$3769. It's sold out, and the planet struggles with carbon emissions.

Thus 80 bit of security can not blindly be said good enough today. 80-bit may be very safe, or clearly not enough, depending on a variety of factors;

Value of the target
Time frame during which a break makes sense
How easier (or harder) it is to check a key compared to SHA-256, due to

use of entropy stretching,
the amount of memory necessary per search engine


Possibility of multi-target attacks, where the attacker's goal is to find any one in several target keys, and they manage to make the cost of testing a candidate key against all target keys mostly independent of their number. That's sometime straightforward (e.g. when a ciphertext block for the same known plaintext block is available block-enciphered under all the target keys).


Addition: I second everything in Thomas Pornin's answer (Feb. 2014), except perhaps &quot;Right now, 80 bits still seem quite sturdy&quot; which is only valid in some contexts, including the context of the question, at least at the time.
In the context intended for KATAN (really low-end devices such as RFID tags), an expectable consequence of key recovery is to be able to clone one single RFID tag. Recovering the key is likely feasible at far lower cost than brute force, by side channel attack or micro-probing, regardless of key size (even though testing a KATAN key requires significantly less work than performing one SHA-256). So right now (Nov. 2015), 80 bits of key is not a major weakness for an individual RFID tag key on rational economic grounds (if we discount the irrationally devastating effect of the announce of any break, and the more rational desire to not have to argue on key size if such break occurs). 80 bits would not be enough for the master key from which the RFID tag keys are derived, or some other uses of KATAN.