From the map, the two shortest borders listed:
1. PENON DE VELEZ DE LA GOMERA
74 m / 242 ft | SPAIN-MOROCCO
Originally a pirate island conquered and fortified by Spain in 1508. It got the world's shortest border in 1934 when a sandstorm left a 74 meters wide isthmus connecting it to the mainland. The peninsula houses a Spanish army base manned by 60 soldiers and is accessed by helicopter.
157 m / 515 ft | BOTSWANA-ZAMBIA
This almost-quadripoint on the Zambezi River is intercepted by a tiny section of Botswana-Zambia border. Historically a busy crossing point, the border section was first defined in 2010 as preparations for a $260M bridge connecting Botswana and Zambia, scheduled to start construction in 2013.