Highbury… afforded her no equals. The Woodhouses were first in consequence there. All looked up to them. She had many acquaintance in the place… but not one among them who could be accepted in lieu of Miss Taylor for even half a day. (1)

I don’t wish to claim expert status but Stephanie I found your question interesting so I will attempt some explanation mostly based on the novels but also some posts in the L&T archives. I think “half a day” may standout as a significant piece of time because of the ritual of polite morning visits that we see in all of JA’s novels. From my reading at L&T the Georgian morning was from breakfast, 9 or 10 AM, until dinner which could be as early as 3PM or as late as 6:30 PM depending of the fashionableness of the household. I think circumstance and degree of friendship often dictated the length of a visit. A half-hour seems to have usually been the shortest time considered polite and from there adding quarter-hour increments. A half-hour seems to be the norm while some, courting gentlemen for example, aim for an hour or so:

In a few days Mr. Bingley returned Mr. Bennet's visit, and sat about ten minutes with him in his library. (P&P, 3)
After sitting with them [Tilney’s] a quarter of an hour, she [Catherine] rose to take leave… (NA, 13)
…for of the quarter of an hour bestowed on Berkeley Street, she [Fanny D] sat at least seven minutes and a half in silence. (S&S, 34)
Four weeks passed away…the shortness of her [Caroline’s] stay…the alteration of her manner… (P&P, 26)
…he had [Henry] spent half an hour with his sister [Mary] the evening before his leaving London… (MP, 41)
Wentworth made a very early return to Mr. Musgrove's civility, and she [Anne] was all but calling there in the same half-hour. (P, 7)
To the Great House accordingly they [Ann & Mary] went… The half-hour was chatted away pleasantly enough (P, 5)
After staying with them half an hour, he [John Dashwood] asked Elinor to walk with him to Conduit Street… (S&S, 33)
Some doubts always lingered in her [Lucy’s] mind… which could only be removed by another half hour's discourse… (S&S, 50)
Call it gossip, if you will, but when Nurse Rooke has half an hour's leisure to bestow on me [Mrs. Smith]… (P, 17)
Their visitors [Georgiana, Darcy, Bingley] staid with them [Lizzy, Gardiners] above half an hour… (P&P, 44)
…he [Sir Thomas] had hoped better things; he had thought that an hour's entreaty from a young man like Crawford… (MP, 33)
"She [Nurse Rooke] sat an hour with me [Mrs. Smith] on Monday evening, and gave me the whole history." (P, 21)
…obliged [Fanny] …to be fixed in their drawing–room for an hour… “Another quarter of an hour,” said Miss Crawford… (MP, 22)
He [Bingley] sat with them [Bennet ladies] above an hour, and was in remarkably good spirits. (P&P, 55)
[Henry] had been sitting with Lady Bertram and Fanny… “Sitting with them an hour and a half!” exclaimed Mary. (MP, 30)

Not all visiting was confined by time. Sometimes duty and friendship called for something more. Here are two examples from the novels, there may be more. In Persuasion Anne gives hours to go out with Lady Russell in Bath: “The following morning Anne was out with her friend [Lady Russell], and for the first hour…” (P, 19). Is this something like going to the mall with friends? In S&S duty comes into play. The Middletons invite Elinor & Marianne to spend not half a day but the “whole of every day in Conduit Street” (S&S, 36) while Mrs. Jennings gives her hours to Charlotte. I think Emma befriends Harriet and quickly establishes a friendly intimacy:

Harriet Smith's intimacy at Hartfield was soon a settled thing. Quick and decided in her ways, Emma lost no time in inviting, encouraging, and telling her to come very often; and as their acquaintance increased, so did their satisfaction in each other. As a walking companion, Emma had very early foreseen how useful she might find her. In that respect Mrs. Weston's loss had been important. …and a Harriet Smith, therefore, one whom she could summon at any time to a walk, would be a valuable addition to her privileges. (4)

Harriet is invited to Hartfield and is encouraged to pop-in often. Emma wishes to fill the void of a friend in whose company she probably spent almost all her waking hours so I can see why she would encourage Harriet to visit for hours. There may be one example of a longer visit in the chapters this week. In Ch. 6 Harriet seems to be wandering Hartfield at will which suggests to me some comfort in her surroundings and that she has been there some amount of time. Emma, Harriet and Mr. Elton are a Hartfield. Harriet was “on the point of leaving the room" when Emma accosts her about having ever sat for a portrait. I don’t know why Harriet leaves Emma and Mr. Elton alone. Maybe she needs to refresh herself or perhaps she only needs something from her coat or cloak pocket however whatever Harriet’s purpose her leaving still strikes me as odd for a typical quarter or half-hour visit. Harriet “was soon back again” but afterwards it would have taken some time to look over all of Emma’s portraits, choose the style of Harriet’s portrait and then have the first sitting. I feel like this could have been a visit lasting half a day. (:D)

I read these L&T posts in preparing this one but before reading please note 43343, 43317 and the links in 49123 contain spoilers for Emma:

Visits in JA's works. Written by Mandy N (January 31, 2006)
http://www.pemberley.com/bin/archives/regarc.pl?read=43343

Morning visits. Written by JulieW (September 21, 2007)
http://www.pemberley.com/bin/archives/regarc.pl?read=48261

Meal times etc Written by JulieW (February 1, 2008)
http://www.pemberley.com/bin/archives/regarc.pl?read=49123