Yarışta beşinci oldum.

Because -ta / -te is the locative case ending, often meaning in, at, during.

• yarış = race
• yarışta = in the race / at the race

So Yarışta beşinci oldum literally means something like I became fifth in the race, which in natural English is I came fifth in the race or I finished fifth in the race.

The ending appears as -ta here because of consonant harmony:

• yarış ends in the voiceless consonant ş
• so -da / -de becomes -ta / -te

Beşinci means fifth.

It is an ordinal number:

• beş = five
• beşinci = fifth

Turkish forms ordinal numbers with endings such as -inci, -ıncı, -üncü, -uncu, depending on vowel harmony.

Examples:

• birinci = first
• ikinci = second
• üçüncü = third
• dördüncü = fourth
• beşinci = fifth

So here, beşinci describes the speaker’s position in the race.

Yes, literally it does.

• olmak = to become / to be
• oldum = I became

In Turkish, ranking expressions often use olmak:

• birinci oldum = I came first
• ikinci oldum = I came second
• beşinci oldum = I came fifth

English usually says I came fifth or I finished fifth, but Turkish commonly says I became fifth. It sounds completely natural in Turkish.

Oldum is in the simple past tense.

Breakdown:

• ol- = verb stem from olmak
• -du / -dı / -dü / -di = past tense marker
• -m = I

Because of sound changes, it appears as oldum rather than a more mechanical-looking ol-dum.

So:

• oldum = I became / I was

In this sentence, it means I came fifth or I finished fifth.

Because Turkish often drops subject pronouns when the verb already shows who the subject is.

• oldum already means I became
• so ben is not necessary

You could say:

• Ben yarışta beşinci oldum.

But usually ben is omitted unless you want emphasis, contrast, or clarity.

For example:

• O birinci oldu, ben beşinci oldum. = He/She came first, I came fifth.

Yes, Turkish word order is flexible, but this order is very natural.

Here:

• Yarışta = in the race
• beşinci = fifth
• oldum = I became

A common neutral pattern in Turkish is to put the verb at the end.

This sentence highlights the situation first, then the result:

• Yarışta beşinci oldum.

Other possible orders can occur, depending on emphasis:

• Beşinci oldum yarışta. → more marked, less neutral
• Ben yarışta beşinci oldum. → adds emphasis to I
• Beşinci ben oldum. → means something more like I was the one who came fifth

So the given version is the most straightforward and natural one.

No, not if you mean I came fifth.

• beş = five
• beşinci = fifth

For rankings, you need the ordinal number, not the cardinal number.

So:

• beş oldum = I became five → not correct here
• beşinci oldum = I became fifth → correct

Yarış often means race or competition/contest, depending on context.

It can refer to:

• a running race
• a car race
• a general competition
• sometimes even a figurative race

So Yarışta beşinci oldum could mean:

• I came fifth in the race
• I placed fifth in the competition

The exact English translation depends on the context already known to the learner.

Yes, but olmak is the standard and most natural choice for stating placement.

For example:

• Yarışı beşinci bitirdim. = I finished the race in fifth place.
• Yarışta beşinci sırayı aldım. = I got fifth place in the race.

But Yarışta beşinci oldum is very common, natural, and simple.

So if you are learning the basic pattern for rankings, this is a very useful one:

• birinci oldum
• ikinci oldum
• üçüncü oldum
• beşinci oldum

They are different cases, so they mean different things.

• yarışta = in the race / at the race
  This is the locative case.

• yarışı = the race as a definite direct object
  This is the accusative case.

Compare:

• Yarışta beşinci oldum. = I came fifth in the race.

• Yarışı kazandım. = I won the race.

So in your sentence, yarışta is used because the race is the setting or context, not the direct object of the verb.

No, not in this sentence.

Here beşinci functions like a predicate expression with oldum:

• beşinci oldum = I became fifth

Because it is not directly modifying a noun like place or position, it does not need an extra noun after it.

Turkish often leaves that noun understood:

• beşinci oldum literally I became fifth understood as I became fifth place / fifth in rank

If you wanted to make it more explicit, you could say something like:

• Yarışta beşinci sırada oldum.
• Yarışta beşinci sırayı aldım.

But the original sentence is already complete and natural.