What is the gauge of wire used in a typical commercial 40 pin IDE cable?
26 or 28 AWG for typical ribbon cables (like the one in the O.P. picture). Ribbons with bigger wires (smaller AWG) are more of a specialty. Here's a datasheet for your generic ribbon cable, which specifies the wire gauge (bottom of the page).
P.S. Of course, if one wire in the ribbon can't carry enough current, you could use several in parallel.
The wire is usually stranded copper wire, usually either 0.32, 0.20, or 0.13 mm^2 (22, 24, or 26 AWG).
One of the most popular sizes of ribbon cable employs 26AWG wire.
In my defense, the latter has a "citation needed" tag. :)
The stuff I use (from www.reichelt.de, this is the 10-wire version):
Flachbandkabel AWG28, 10-pol., grau, 30m-Rolle
• Typ: Flachbandkabel • Rastermaß: 1,27 mm • Aderanzahl: 10pol • Kabelquerschnitt: 0,09 mm² • Verpackung: Rollenware • Kabellänge: 30,5 m • Farbe: grau • Gewicht:0,61 kg