This KBC4059 QSL card for Dick "Road Runner" Jalbert of Agawam, Massachusetts, is slightly larger than a business card.
Eyeball QSLs are, to my understanding, used for the exchange of ham-radio contact information at face-to-face gatherings (as opposed to being sent through the mail, as standard-size QSL cards often are). Eyeball QSLs might be handed out at club meetings or given to prospective members or newcomers to the hobby.
On the blog titled VU2SGW, a ham-radio hobbyist from India wrote the following in 2009:
"The term eyeball in HAM lingo means nothing but meeting a person and having eye to eye contact. Eyeball QSL Cards can make a lasting impression at club meetings, hamfests, or anywhere hams gather. They are visiting card size and are easier to carry around than a QSL Card. When I happen to run across a fellow Ham in public, I don’t miss the opportunity to hand out my eyeball card. I do digital graphic designing as an hobby, it was an added benefit. I have designed some custom made eyeball for me and my friends as well."
Meanwhile, on the GL1800Riders message board in 2012, a poster nicknamed Halfling from Murfreesboro, Tennessee, began writing about eyeball QSLs and potential meetups among hobbyists. Read his two message-board threads here and here.
In August 2013, I featured a 1960s Errol Engraving advertisement for ham-radio supplies. On of the items available was 60 self-sticking "Eye Ball" QSL labels for 50 cents.
Check out all of the vintage QSL cards I've posted on Papergreat.