How to rebuild the Temple, even though most of us are ritually impure

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Rambam {Maimonides} end of Hilchot Beit Habechira Chapter 7 as translated by Chabad
http://www.chabad.org/library/article_cdo/aid/1007200/jewish/Beit-Habechirah-Chapter-7.htm

There was a place in the upper storey [of the Temple]78which was located directly opposite the Holy of Holies.79 It was entered only once in seven years, to [inspect it] and find out what is necessary for its repair.80

When builders [are required] to enter the Temple building to construct or repair it, or to remove an impure object,81 it is a mitzvah for the [craftsmen] who enter to be priests who do not possess any disqualifying physical deformities.82

If no [capable craftsmen meeting those criteria] can be found, priests with disqualifying deformities should enter.83

If none are found, Levites should enter.84

If none are found, Israelites should enter.

It is a mitzvah for [those who enter] to be ritually pure. If no [capable craftsman] who are ritually pure can be found, impure [craftsmen] may enter.85

[If there is a choice between a craftsman] who is impure and a priest with a disqualifying deformity, the priest with the deformity should enter, for [although the prohibitions against] ritual impurity are put aside in regard to matters which concern the entire people, [they are not relaxed completely].86

All those who enter to repair the Temple87 should be lowered down inside crates [from the upper floor].88 If no crates are available or if it is impossible [to make arrangements for them to enter] using crates, they may enter through the [usual] entrances.
Footnotes by Chabad Translator
78.   
See Chapter 4, Halachah 13.
79.   
The Tosefta (Kelim 1:7) states:
Abba Saul declares: "[The status of] the upper storey of the Holy of Holies is more strict [than that of] the Holy of Holies. In regard to the Holy of Holies, the High Priest enters four times each year, on Yom Kippur...[In contrast,] they would only enter the upper storey of the Holy of Holies once in seven years..."
They replied to him: "That is not considered a distinguishing quality."
Certain commentaries explain that the Rambam placed this Halachah here to indicate his acceptance of Abba Saul's view. Although there are other reasons to support this argument, it would appear that the Rambam subscribes to the other view mentioned in the Tosefta. In Halachah 13, he stated that there are ten levels of holiness and proceeded to enumerate them, concluding with the highest level in Halachah 22.
Furthermore, the opening phrase of each of those halachot, states: "... is holier than it..." and this halachah does not begin in that fashion.
80.   
Pesachim 86a mentions three opinions concerning the frequency in which this chamber was entered: twice in seven years, once in fifty years, and once every seven years as quoted above. See Tosafot Yom Tov, Middot 4:5.
81.   
Eruvin 104b and 105a explain that this impurity could be either the body of one of the eight crawling species (sheratzim) which convey ritual impurity, or alternatively, sacraments belonging to idol worship that were placed in the Temple. II Chronicles (Chapter 29) explains that the Temple was cleansed of idol worship by King Chezekiah, and that narrative serves as a source from which these laws were derived.
82.   
As mentioned in Halachah 20, priests with disqualifying physical deformities were generally forbidden to proceed beyond the Priestly Courtyard.
83.   
Eruvin, loc. cit., explains that the prohibition against these priests entering the more sanctified portions of the Temple implies the permission to enter should they be required for their craftmanship.
84.   
The Levites are granted precedence, since they are allowed to proceed beyond the Israelites and to stand on the steps within the Priestly Courtyard.
85.   
This leniency is allowed because the restrictions stemming from the laws of ritual purity do not apply to cases involving communal offerings for the entire Jewish people.
86.   
This statement has stirred much controversy among the commentaries. See the Kessef Mishneh and the Mishneh LiMelech.
Although the Sages all agreed that the restrictions stemming from ritual impurity did not apply to cases involving communal offerings for the entire Jewish people, they debated (Yoma 6b) the nature of the leniency. Rav Nachman maintains that once a need arises, all the restrictions governing ritual impurity are relaxed entirely. Rav Sheshet argues that these restrictions are only "put aside" when there is no alternative, and wherever possible, the restrictions should be maintained. In Hilchot Bi'at HaMikdash 4:15, the Rambam rules according to Rav Sheshet's opinion.
In our mishnah, both a craftsman who is impure, and a priest with a disqualifying physical deformity, are prohibited from entering the Temple. The prohibition against entering while impure is more severe. Thus, Rav Sheshet's opinion would explain that although the restrictions against entering while impure are "put aside" to repair the Temple, total license is not granted. Hence, in this instance, since an alternative exists, it should be employed, and the priests with the deformities should be allowed to enter.
87.   
Middot 4:5 mentions this practice only in regard to the Holy of Holies. However, Eruvin (loc. cit.) and the Tosefta (loc. cit.) apply the concept to the entire Temple building.
88.   
Middot 4:5, and Rambam (Chapter 4, Halachah 13) explain that this practice was instituted "so that they would not satiate their eyes, [gazing at] the chamber of the Holy of Holies."

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Rabbi Hillel Ben Shlomo raises the suggestion that Moses Montefiore went up to the Temple Mount to the more sanctified areas based on the above halacha in the Rambam in order to try to negotiate with the Arabs, the right to put an altar on the Mount and offer sacrifices. Because he thought that publicly revealing his intentions would ruin the negotiations, he gave other explanations why he went to those areas and did not tell his rabbinic critics what was his true intentions.

see the Hebrew article at http://the--temple.blogspot.co.il/2014/08/blog-post_8.html

I am just reporting what Rabbi Hillel Ben Shlomo wrote. I did not check out the history sufficiently to voice an opinion.