Eastman is a product of the Ipswich Town Academy, where he progressed through the ranks since the age of eight before making his first-team debut in 2010, a match in which he was sent off for a second bookable offence.

After making his Colchester debut in August 2011, he joined Crawley Town in an emergency loan deal in September 2011, where he made six appearances, before returning to the U's and establishing himself as a regular starter in the centre of defence.

[13] He was an unused substitute for the Ipswich first-team matches against Nottingham Forest and Derby County in April 2010, before being offered a professional contract by the club.

[20] In April 2011, alongside teammates Jack Ainsley and Conor Hourihane, Eastman was offered a six-month contract, with his deal set to expire in the summer.

[23] Eastman rejected the offer of a six-month contract from Ipswich and instead decided to join local rivals Colchester United on a two-year deal on 19 May 2011 on a free transfer.

[24][25] He made his debut for the club in the Football League Trophy on 30 August 2011, replacing Matt Heath with seven minutes remaining of their 3–1 home defeat to Barnet.

[26] On 13 September, Eastman signed for Crawley Town in an emergency one-month loan deal after an injury crisis struck the League Two club.

[33] Manager John Ward then confirmed that Eastman would keep his place in the starting line-up on the back of his impressive performance and with Matt Heath still sidelined through injury.

[38] Eastman started his second season with Colchester partnering Magnus Okuonghae in the centre of defence in their season-opening League Cup 3–0 defeat to Yeovil Town on 14 August 2012.

[39] He scored his first goal of the campaign on 20 November when Colchester were defeated 3–1 at home to Coventry City, heading in Anthony Wordsworth's 71st minute free-kick.

[41] He set himself the target of continuing to hold down a regular starting position in Joe Dunne's squad, despite strong competition for places in the team.

[51] Lining up alongside Okuonghae for his fourth season with the U's, Eastman featured in Colchester's 2–2 opening day home draw with Oldham Athletic on 9 August 2014.

[59] Eastman was involved in a sickening clash of heads with fellow Colchester defender Alex Wynter on 30 January in the U's FA Cup fourth round defeat by Tottenham Hotspur.

After just two-minutes of play, the two collided, leaving Wynter unconscious and suffering a seizure on the field, and Eastman drawing blood and requiring treatment off the pitch.

[65] Eastman made a goalscoring return to action in the 2016–17 season when he scored Colchester's goal in their 1–1 opening day draw with Hartlepool United at Victoria Park.

[68] Following an injury sustained during Colchester's 1–1 draw with Blackpool on 4 February 2017, it was reported that Eastman would be ruled out of action for the remainder of the season after requiring an operation on ruptured ankle ligaments.

Released by Colchester United upon expiration of his latest contract in the summer of 2023, Eastman had a permanent offer from Harrogate Town but turned it down,[82] citing a desire to remain close to his home in Essex.