Is there a limit on how much current a pin can sink? I've been experimenting with an 8x8 LED matrix controlled by an Arduino Uno. As I think is typical, the matrix uses a common anode for each row, and a common cathode for each column. 

At the moment, I've got all the matrix pins connected directly to IO pins on the Uno, and I haven't had any problems lighting the LEDs one-by-one. Going through the whole matrix like this makes updates a little slow though, meaning the LEDs aren't as bright as I'd like.

As far as I know, I can't safely light up an entire row at once, because an individual pin on the Uno isn't capable of sourcing enough current to drive 8 separate LEDs at the same time (requiring at least 10mA each to be bright enough).

It occurred to me that the reverse might be safer. If I light up an entire column at a time, then each pin only has to source enough current for one LED, which should be no problem. However, it relies on one pin potentially sinking the current from all 8, totalling at least 80mA.

Is this possible, or am I going to fry my board?
am I going to fry my board?


Pretty likely outcome. Here's why:

The Arduino Uno uses the atMEGA328 microcontroller, which has an absolute maximum rating of 40 mA source or sink per GPIO. Also, the total current through the supply or ground rails (i.e. the total of all current OP wants the GPIO pins to sink, or source) is rated to a maximum of 150 200 mA. 

Realistically, the current graphs in the atMEGA328 datasheet do not even go all the way to 40 mA, hence it is safest to restrict the current to perhaps 20 mA total per GPIO.

In other words, whether a GPIO is used as source or sink, the limits remain pretty low, and in practice, should be kept much lower than the absolute maximum ratings. Even below absolute maximum, running a microcontroller to too high a temperature reduces its life expectancy.

Solution: 

Use either BJTs, MOSFETs, or some driver IC such as the ULN2003 to actually drive the current through the LEDs.