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A person with ambitions or aspirations that far exceed their means or abilities would be quixotic. The term is a good fit because of its definition and because it brings to mind Don Quixote providing a mental picture of aspirations exceeding abilities.

quixotic

ADJECTIVE

Extremely idealistic; unrealistic and impractical.

‘a vast and perhaps quixotic project’ English Oxford Living Dictionaries

In this case an example would be: "Bob has quixotic ambitions"

Etymology

Quixotic Has Roots in Literature

If you guessed that quixotic has something to do with Don Quixote, you're absolutely right. The hero of the 17th-century Spanish novel El Ingenioso Hidalgo Don Quijote de la Mancha (by Miguel de Cervantes) didn't change the world by tilting at windmills, but he did leave a linguistic legacy in English. The adjective quixotic is based on his name and has been used to describe unrealistic idealists since at least the early 18th century. Merriam-Webster

You could work in an allusion to Thurber's character Walter Mitty.

Wikipedia explains:

Walter Mitty is a fictional character in James Thurber's first short
story "The Secret Life of Walter Mitty", first published in The New
Yorker on March 18, 1939, and in book form in My World and Welcome
to It in 1942. Thurber loosely based the character, a daydreamer, on
his friend Walter Mithoff. It was made into a film in 1947 starring
Danny Kaye, with a remake directed by, and starring Ben Stiller
released in 2013.The character's name has come into more general use
to refer to an ineffectual dreamer and appears in several
dictionaries. The American Heritage Dictionary [see also the American Heritage Dictionary of Idioms] defines 'a Walter
Mitty' as "an ordinary often ineffectual person who indulges in
fantastic daydreams of personal triumphs".

Examples:

He has been described as the Walter Mitty of the political world, a
complete nobody who has somehow contrived a career out of standing on
a soapbox and protesting against anything the mainstream politicians
do.

...............

My father worked for the same company for over 50 years and never even
left his home state, but he was always something of a Walter Mitty,
dreaming about a life of adventure.

[both Farlex Dictionary of Idioms]

Wannabe (mild slang/ informal) might fit a person aspiring to enter some other social group. There's a strong implication by the person using the term that he does not think the aspirant will ever succeed. Use as a noun ("a pathetic wannabe") or adjective ("a wannabe rock-star")

https://www.merriam-webster.com/dictionary/wannabe

If he is actively pursuing those unrealistic ambitions, you could say that he is biting off more than he can chew.

One could say that "their ego is writing checks their body can't cash".

From the 1986 movie Top Gun (imdb):

They're a dreamer

dream·er (n)

a person who dreams or is dreaming.

a person who is unpractical or idealistic. "a rebellious young dreamer"

synonyms: fantasist · fantasizer · daydreamer · romantic · sentimental(ist)

Your exchange with @Edwin Ashworth makes it seem more along the lines of egocentric (disconnect from reality). As far as hopeless, you could describe them or the situation as classically pathetic.

path·et·ic (adj)

marked by sorrow or melancholy : sad

pitifully inferior or inadequate "the restaurant's pathetic service"

Edit:

You could also say they bit off more than they could chew or they are out of their league. Agree with @Robby Cornelissen and @Lee Daniel Crocker

Overreach is a somewhat bland term for this. Collins:

If you say that someone overreaches themselves, you mean that they
fail at something because they are trying to do more than they are
able to.

One might say the person is "Too Big for (his/her) Britches"

Urban Dictionary

Adjective phrase meaning that your assumed position is slightly larger than the actual position you belong in, hence the idiom referring to the too big for the pair of pants.

When you smarted off to the boss yesterday, everyone in the office thought you were too big for your britches.

A phrase I have often heard in that context is Head in the clouds.

Head in the clouds

- (plural heads in the clouds) (figuratively, usually with have or with) Used to indicate that a person has fantastic or impractical dreams

There is a term in the psychological literature that might apply here or could be adapted for your use. The Dunning-Kruger effect is a cognitive bias in which a person believes their abilities are stronger than they actually are. See the wikipedia entry here for more information:
https://en.wikipedia.org/wiki/Dunning%E2%80%93Kruger_effect

This term seems to have the "disconnect from reality" emphasis that you mentioned you were looking for in the comments too.

For example, you might write something such as:

The candidate, having years of experience in politics believed she had
the election won. She was certain she was better able to connect
emotionally with voters, but when the results were tallied, she had
lost to a political neophyte. The Dunning-Kruger effect, it seems, had robbed
her of the election.

(Personal writing)

As empirical evidence of meta-ignorance, I describe the
Dunning–Kruger effect, in which poor performers in many social and intellectual domains seem largely unaware of just how deficient their
expertise is. Their deficits leave them with a double burden—not only
does their incomplete and misguided knowledge lead them to make
mistakes but those exact same deficits also prevent them from
recognizing when they are making mistakes and other people choosing
more wisely.

Dunning, David. Chapter five - The Dunning–Kruger Effect: On Being Ignorant of One's Own Ignorance. Advances in Experimental Social Psychology. V44 (2011), p247-96.

You could say that his plans or ambitions are "grandiose". For example, the artilleryman in The War of the Worlds "briefly persuades [the narrator] of a grandiose plan to rebuild civilization by living underground".

Fanciful

You may want to use the word "fanciful" in the context of "fanciful about his/her aspirations."

The definition of fanciful means that something is "overimaginative and unrealistic."

See here and here

I think megalomaniac comes close to what you are referring to:

someone who has an unnaturally strong wish for power and control, or thinks that they are much more important and powerful than they really are.

(Cambridge Dictionary)

I have heard the term " Would be could be " This is somebody that would be great if he could be but he really cannot be anything other than mediocre .