A short summary since the answer became a bit long
- House elves magic seems to be unique and bound to their nature. It is most probably not possible for a wizards to use/learn this magic.
- Goblin magic seems to be very similar to wizard magic but they keep it secret. Also they have developed the skills they use for a long time and it won't be easy for a single wizard perform the same magic.
Now the long story with some examples:
The magic of House Elves is described as "different" without providing further direct information about it. Also they don't need wands at all although it is not clear whether they can use wands but are not allowed to or they just don't need them. As noted in a comment to the previous answer - house elves are capable of using wands because Winky was accused of casting the dark mark spell with the wand of Harry (Harry Potter and the Goblet of Fire)
Here an example from Harry Potter and the Deathly Hallows, chapter 10:
"How did you get away?" Harry asked, and he was not surprised to hear
Kreacher raised his ugly head and looked Harry with his great,
"Master Regulus told Kreacher to come back," he said.
"I know ... but how did you escape the Inferi?"
Kreacher did not seem to understand.
"Master Regulus told Kreacher to come back," he repeated.
"I know, but ..."
"Well, it's obvious, isn't it, Harry?" said Ron. "He Disapparated!"
"But ... you couldn't Apparate in and out of that cave," said Harry,
"otherwise Dumbledore ..."
"Elf magic isn't like wizard's magic, is it?" said Ron, "I mean, they
can Apparate and Disapparate in and out of Hogwarts when we can't."
Also in chapter 23 noone was able to disapparate outside of Malfoy's cellar
Ron was now trying to Disapparate without a wand.
"There's no way out, Ron," said Luna, watching his fruitless efforts.
"The cellar is completely escape-proof. I tried, at first. Mr.
Ollivander has been here for a long time, he's tried everything."
A bit later Dobby comes and saves everyone:
Harry hit Ron on the arm to stop him shouting, and Ron looked
terrified at his mistake. Footsteps crossed the ceiling overhead:
Draco marching Griphook to Bellatrix.
Dobby's enormous, tennis-ball shaped eyes were wide; he was trembling
from his feet to the tips of his ears. He was back in the home of his
old masters, and it was clear that he was petrified.
"Harry Potter," he squeaked in the tiniest quiver of a voice, "Dobby
has come to rescue you."
"But how did you?"
An awful scream drowned Harry's words: Hermione was being tortured
again. He cut to the essentials.
"You can disapparate out of this cellar?" he asked Dobby, who nodded,
his ears flapping.
"And you can take humans with you?"
Dobby nodded again.
So most probably the magic of house elves cannot be repeated by wizards since its source is the very nature of the elves and their deep connection with their masters.
This looks different about Goblins. Their magic seems to be similar to the one of the wizards but they keep it secret.
"Well, goblins can do magic without wands," said Ron.
"That is immaterial! Wizards refuse to share the secrets of wand-lore
with other magical beings, they deny us the possibility of extending
"Well, goblins won't share any of their magic either," said Ron. "You
won't tell us how to make swords and armor the way you do. Goblins
know how to work metal in a way wizards have never "
This implies that:
- Goblins can use wands but don't have the skills to craft them on their own. Most probably wizard craftsmen just refuse to sell wands to goblins since we don't see any goblins with wands.
- Wizards could eventually reproduce the work of goblins but lack the knowledge (which the goblins keep secret) and the skills.
Additionally we see lots of magic used to protect the vaults in the bank. However it is not clear whether the goblins cast the spells on their own or employ the services of some wizards (like Bill Weasley).
Goblins seem to have some unique (to the wizzarding world) "muggle-like" skills. They are adept at finances since Gringotts holds absolute monopoly in banking in the wizard world. This however might have also an out-of-universe explanation - a second bank did not fit in the plot and JKR did not bother describing it.