The notable object is ...
... the Rosetta Stone
This crossword is "incomprehensible", because ...
... the across answers seem off. We can find reasonable words for the down entries, but the across entries have two word lengths: One in the grid and one in brackets after the clue. Clearly, there's something going on.
The hint ...
... is useful and is a cryptic clue. Narushiteli has solved it in a comment to a now deleted answer.
Babelfish = BABE (newborn?) + L(ady) + anagram of (HIS F(ather's))
The Babelfish is an animal from The Hitchiker's Guide to the Galaxy, which one can put in one's ear and have it translate everything. (Sneaky way to get rid of subtitles.) Babelfish was also the nme of one of the first online translators. (I believe it was a precursor to the boringly named Google Translate.)
We can start with ...
... the across answers. The numbers in brackets provide the length:
1. Clattering or banging, for example 5 NOISE
2. Has never been done before 8 ORIGINAL
3. Fiery pit 4 HELL
4. Exam solution 8 ANSWER
5. Very cream of the crop 4 BEST
6. Don't lose this while traveling 7 LUGGAGE
7. Fruit or vegetable quality 7 ORGANIC
8. To use as an alternative 7 IMPLACE
9. Immediate successor 4 NEXT
10. Podcast invitee 5 GUEST
11. Heavy reading 10 LITERATURE
12. -a-Rm joint 5 ELBOW
13. Provider 8 SUPPLIER
The first letters of these answers spell No hablo Ingles, which is "I don't speak English" in Spanish. (Stiv has found the acrostic.) So we are looking for an answer in another language or other languages.
So ...
... let's see. HELL, for example, is enfer in French. If we enter it in the grid, we see, that the thing needed for an omelet, which surely must be eggs, ends in F. The French word for egg, oeuf, conveniently ends in an F. But the answers aren't all in French.
Each numbered across and down clue is written in another language. We can find out which by looking at the first letters of the across and down clues. No. 3 is "Fiery pit" and "Required to make an omelet", which gives fr = France, where French is spoken. (The codes are not ISO language codes, but country codes; we have va for Vatican, which is used for Latin.)
(I'll ignore all ligatures and diacritics from now on and demote all letters to their Latin base form, even if that means that I'll have to write "Gepack", which I can't do lightly.)
Now, the across answers are:
1. Clattering or banging, for example 5 NOISE 4 HLUK (cz)
2. Has never been done before 8 ORIGINAL 8 IZVORNIK (hr)
3. Fiery pit 4 HELL 5 ENFER (fr)
4. Exam solution 8 ANSWER 9 RESPUESTA (es)
5. Very cream of the crop 4 BEST 7 OPTIMUM (va)
6. Don't lose this while traveling 7 LUGGAGE 6 GEPACK (de)
7. Fruit or vegetable quality 7 ORGANIC 15 LUONNONMUKAINEN (fi)
8. To use as an alternative 7 IMPLACE 6 YERINI (tr)
9. Immediate successor 4 NEXT 8 PROSSIMO (it)
10. Podcast invitee 5 GUEST 7 HOSPEDE (pt)
11. Heavy reading 10 LITERATURE 8 IRODALOM (hu)
12. -a-Rm joint 5 ELBOW 3 COT (ro)
13. Provider 8 SUPPLIER 8 DOSTAWCA (pl)
The down clues ...
... can now be solved. (Thanks to richardb, who filled in the last blanks in comments.)
1. Zeal 4 ELAN ELAN (cz)
2. Requests to god 7 PRAYER MOLITVA (hr)
3. Required to make an omelet 4 EGG OEUF (fr)
4. Sides of a house 5 WALLS TAPIA (es)
5. Active and lively 7 VIVID IMPIGER (va)
6. Early form of the piano 7 HARPSICHORD CEMBALO (de)
7. Inhuman, but comes in many ... 5 BEAST ELAIN (fi)
8. Readiness for potential problem 6 CAUTION TEDBIR (tr)
9. Twelve that decide the fate of man 6 JURY GIURIA (it)
10. Temple 9 RELIQUARY RELICARIO (pt)
11. Unit of precipitation 8 RAINFALL ESOCSEPP (hu)
12. Open framework used to support ... 8 SCAFFOLD ESTACADA (ro)
13. Long-term artillery barrage 8 CANNONADE KANONADA (pl)
The completed grid ...
E
Y E R I N I H L U K
L M A
G E P A C K I Z V O R N I K R
I I L E
L U O N N O N M U K A I N E N L
R T I
I O V C O T
A R E S P U E S T A A
U A I R C
K E N F E R O P T I M U M I E
A S I P R O S S I M O
N O A E B
D O S T A W C A E G A
N E S H O S P E D E L
A D E T R O
D B P A
A I P C
R I R O D A L O M
D
A
The grid entries ...
... are another acrostic. In the order they appear in the list of clues, they spell out Hieroglyphic Demotic et Greek.
This is a reference to the Rosetta Stone, a stele found near the Egyptian city of Memphis. It has nearly identical versions of the same text in Egyptian Hieroglyphic srcipt, in Egyptian Demotic script and in Greek, which made it possible to decipher the Egyptian scripts.
Well, let's just say that Google Translate is a modern version of the Rosetta stone. (And a primitive form ob Babelfish.)