Dostál was born on 22 June 2000 in Brno, Czech Republic,[1] to parents Martin and Renata,[2] but he is a native of Bedřichovice.

[5] Dostál entered the 2018 NHL entry draft as the number one ranked European prospect goaltender and was selected 85th overall by the Anaheim Ducks.

[14] On 25 November 2020, it was announced that the Ducks had exercised their playing rights on Dostál and invited him officially join their organization and move to North America.

[16] Dostál attended the Ducks' training camp in January[17] but was re-assigned to their American Hockey League affiliate, the San Diego Gulls, to start the season.

By doing so at the age of 20, Dostál became the youngest goaltender in franchise history to win his first three starts, passing John Gibson by two years.

[22] Dostál's rookie season involved him becoming more comfortable on the smaller ice surfaces that were standard in North America but much larger in the Liiga.

[30] On 9 January 2022, Dostál made his NHL debut in place of Anthony Stolarz who had played the previous night.

[33] Upon rejoining the Gulls, Dostál recorded his first career AHL shutout on 26 February, against the Henderson Silver Knights.

[35] A few days later, Dostál made 51 saves and scored an empty net goal to clinch a 5–2 win over the Colorado Eagles.

[37] After participating in the Ducks' training camp and preseason ahead of the 2022–23 season, Dostál returned to the AHL for his third campaign with the Gulls.

[40] In December, Dostál became the first Ducks rookie goaltender with multiple 40-save appearances after he made over 40 saves in two games against the Calgary Flames and Edmonton Oilers.

[41] Despite his performance, the Ducks reassigned Dostál to the Gulls on 1 January, after veteran goaltender Anthony Stolarz recovered from an injury and was able to backup starter John Gibson.

[46] Dostál made an immediate positive impact on the Ducks lineup as he led rookie goaltenders with four wins over five appearances in October.

[47] On 3 January 2024, Dostál set a franchise record of 55 saves for the Anaheim Ducks although the team would fall to the Toronto Maple Leafs 2–1 in overtime.

[49] Later that month, Dostál helped the Ducks end their seven-game losing streak by earning his first NHL shutout against the Chicago Blackhawks on 22 March.