Marsh is the first son of Geoff Marsh and elder brother of Mitchell Marsh, both of whom have played for the Australian national side.

Marsh is married to Channel 7 journalist Rebecca O'Donovan, now Rebecca Marsh. They have one child. His brother-in-law is animator and Steam Train co-host Ross O'Donovan. He attended Wesley College in South Perth.

Off the field, Marsh has suffered from occasional lapses of discipline. In November 2007, he was suspended for two matches by the WACA for drinking alcohol to excess with teammate Luke Pomersbach.

In October 2012, during the Champions League Twenty20 tournament in South Africa, Marsh was dropped from the Scorchers last game, along with brother Mitchell, after a night of partying to celebrate Mitchell's 21st birthday.

In 2017, Marsh had an eye test, which showed he was short-sighted, a condition with which he had been batting and fielding for a considerable time. He started wearing contact lenses. This caused surprise in the cricket community, for whom test match contracts normally include sight tests as a routine requirement.

Marsh and his younger brother Mitchell attended Wesley College in Perth where they excelled in cricket. In 1998 Shaun set the record, which was broken ten years later, for the highest average (210) in the Public Schools Association's Darlot Cup cricket competition.

After representing Western Australia at Under 17 and Under 19 level, Marsh represented Australia at both the 1999–2000 and 2001–02 ICC Under-19 World Cups. In the 1999–2000 tournament, held in Sri Lanka, he was the second highest run scorer for Australia behind Shane Watson, and other notable teammates included played alongside Michael Clarke, Nathan Hauritz and Mitchell Johnson. At the 2001–02 tournament in New Zealand, which Australia won, he scored the fourth most runs for the tournament, with the captain of the Australian side, Cameron White being the leading run scorer.

During this time, Marsh made his first-class cricket debut in 2000–01, against South Australia at the WACA Ground in March 2001. He played as a middle order batsman, and played the final three games of the season, before continuing to represent both the Australian Under 19 side and the Australian Cricket Academy throughout 2001 in the lead-up to the World Cup in New Zealand in January to February 2002.