北京奥运会是2008年举办的。
"北京奥运会是2008年举办的" is a [topic + comment/opinion] sentence
(It is obviously wrong, it makes 2008年 the subject. I will state the correct sentence later in this post)
The '的' here is a [possessive marker]. It marks the omitted object to be possessed by [2008年]
2008年 举办的(项目)
2008年 is the subject; 项目 (or other noun refers to 北京奥运会) is the omitted object
The correct sentence should be:
"北京奥运会是在 2008年 举办的" (Beijing Olympics is hosted in 2008)
the subject '北京' is omitted, "北京奥运会是(北京)在 2008年 举办的"
In this sentence, '的' is an [adjective marker]. It turns "在 2008年举办" into an adjective phrase "在 2008年 举办的"
This adjective phrase describes 北京奥运会 (Beijing Olympics that is hosted in 2008)
'的' cannot be omitted in this sentence.
To avoid using 的, you can change this [topic + comment/opinion] sentence into a [SVO] sentence and write "(北京)在 2008年 举办北京奥运会" - "(Beijing) held the Beijing Olympic Games in 2008"
Since we do not omit the subject '北京' here, we can omit the '北京' in '北京奥运会' instead -- "北京在 2008年 举办(北京)奥运会"